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                                                                    EXHIBIT 10.1

                         REGISTRATION RIGHTS AGREEMENT

     This Registration Rights Agreement is dated as of December 13, 1999 by
and between Titan Resources Holdings, Inc., a Delaware corporation (together
with its successors and assigns, the "Company"), and Union Oil Company of
California, a California corporation ("Union Oil").

     1.   Background. The Company has agreed to issue to Union Oil certain
          ----------
shares of the Company's common stock, par value $.01 per share (the "Common
Stock"), pursuant to the transactions contemplated by (i) the Agreement and Plan
of Merger dated as of the date hereof among Union Oil, the Company, TRH, Inc.,
and Titan Exploration Inc., a Delaware corporation (as it may be amended, the
"Merger Agreement") and (ii) the Non-Dilution Agreement dated as of the date
hereof between the Company and Union Oil (as it may be amended, the "Non-
Dilution Agreement").

     2.   Registration under Securities Act, etc.
          --------------------------------------

          2.1  Registration on Request.
               -----------------------

               (a)  Concurrently with or from time to time after the Initial
          Registration Date, upon the written request of any Holder of
          Registrable Securities (a "Requesting Holder"), requesting that the
          Company effect the registration under the Securities Act of all or a
          portion of such Requesting Holder's Registrable Securities and
          specifying the intended method of disposition thereof and whether or
          not such requested registration is to be an underwritten offering,
          subject to the limitations set forth in subsection (e) of this Section
          2.1, the Company will use its best efforts to effect the registration
          under the Securities Act of the Registrable Securities which the
          Company has been so requested to register by the Requesting Holder
          (which request shall specify the intended method of disposition of
          such Registrable Securities), to the extent requisite to permit the
          disposition (in accordance with the intended methods thereof as
          aforesaid) of the Registrable Securities so to be registered (a
          "Demand Registration").

               (b)  Registration of Other Securities. Whenever the Company shall
                    --------------------------------
          effect a Demand Registration pursuant to this Section 2.1 in
          connection with an underwritten offering, no securities other than
          Registrable Securities shall be included among the securities covered
          by such registration unless (i) the managing underwriter of such
          offering shall have advised the Requesting Holder in writing that the
          inclusion of such other securities would not adversely affect such
          offering or (ii) the Requesting Holder shall have consented in writing
          to the inclusion of such other securities.

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               (c)  Registration Statement Form. A Demand Registration under
                    ---------------------------
          this Section 2.1 shall be on such appropriate registration form of the
          Commission (i) as shall be selected by the Company and as shall be
          reasonably acceptable to the Requesting Holder, and (ii) as shall
          permit the disposition of such Registrable Securities in accordance
          with the intended method or methods of disposition specified in the
          request for such registration. The Company agrees to include in any
          such registration statement all information which the Requesting
          Holder shall reasonably request.

               (d)  Expenses. The Company will pay all Registration Expenses in
                    --------
          connection with any Demand Registration requested pursuant to this
          Section 2.1. Selling Expenses in connection with this Section 2.1
          shall be incurred by the Requesting Holder.

               (e)  Limitations on Requested Registrations. The Company's
                    --------------------------------------
          obligation to take or continue any action to effect a Demand
          Registration under this Section 2.1 shall be subject to the following:

                    (i)   The Company shall not be required to effect a Demand
               Registration pursuant to this Section 2.1 unless the Requesting
               Holder has requested the registration of at least the lesser of
               (i) 4,300,000 Registrable Securities, adjusted for any increase
               or decrease in the number of issued shares of the Common Stock
               resulting from a stock split, reverse stock split, stock
               dividend, combination or reclassification of the Common Stock, or
               any other increase or decrease in the number of issued shares of
               Common Stock effected without consideration by the Company or
               (ii) $50 million in Fair Market Value of Registrable Securities.

                    (ii)  The Company shall not be required to effect a Demand
               Registration pursuant to this Section 2.1 more than one time in
               any twelve (12) month period.

               (f)  Selection of Underwriters. If a Demand Registration pursuant
                    -------------------------
          to this Section 2.1 involves an underwritten offering, the underwriter
          or underwriters thereof shall be selected by the Company, with the
          reasonable approval of the Requesting Holder.

          2.2  Incidental Registration.
               -----------------------

               (a)  Right to Include Registrable Securities. If the Company at
                    ---------------------------------------
          any time proposes to register any of its securities under the
          Securities Act (other than (i) in connection with a registration of
          any employee benefit, retirement or similar plan, or (ii) with respect
          to a transaction governed by Rule 145 under the Securities Act, or

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          (iii) pursuant to Section 2.1), whether or not for sale for its own
          account, it will each such time give prompt written notice to the
          Holders of its intention to do so and of the Holders' rights under
          this Section 2.2. Upon the written request of the any Holder made
          within thirty (30) days after the receipt of any such notice (which
          request shall specify the Registrable Securities intended to be
          disposed of and the intended method of disposition thereof), the
          Company will use its best efforts to effect the registration under the
          Securities Act of all Registrable Securities that the Company has been
          so requested to register, to the extent requisite to permit the
          disposition (in accordance with the intended methods thereof as
          aforesaid) of the Registrable Securities so to be registered, provided
                                                                        --------
          that if, at any time after giving written notice of its intention to
          register any securities and prior to the effective date of the
          registration statement filed in connection with such registration, the
          Company shall determine for any reason not to register or to delay
          registration of such securities, the Company may, at its election,
          give written notice of such determination to the Holders participating
          in such registration and, thereupon, (i) in the case of a
          determination not to register, shall be relieved of its obligation to
          register any Registrable Securities in connection with such
          registration (but not from its obligation to pay the Registration
          Expenses in connection therewith), without prejudice, however, to the
          rights of the Holders to request that such registration be effected as
          a Demand Registration under Section 2.1, and (ii) in the case of a
          determination to delay registering, shall be permitted to delay
          registering any Registrable Securities, for the same period as the
          delay in registering such other securities. No registration effected
          under this Section 2.2 shall be deemed to have been effected pursuant
          to Section 2.1 or shall relieve the Company of its obligation to
          effect any registration upon request under Section 2.1. The Company
          will pay all Registration Expenses in connection with each
          registration of Registrable Securities requested pursuant to this
          Section 2.2 and any Selling Expenses shall be incurred by the
          participating Holders.

               (b)  Priority in Incidental Registrations.
                    ------------------------------------

                    (i)   If (A) a registration pursuant to this Section 2.2
               involves an underwritten offering of the securities so being
               registered for sale for the account of a stockholder (other than
               the Holders) exercising a demand registration right pursuant to
               another registration rights agreement to be distributed (on a
               firm commitment basis) by or through one or more underwriters of
               recognized standing under underwriting terms appropriate for such
               a transaction, and (B) the managing underwriter of such
               underwritten offering informs the Company and the Holders in
               writing of its belief that the number of securities requested to
               be included in such registration exceeds the number that can be
               sold in (or during the time of) such offering, then the Company
               will include in such registration, to the extent of the number
               that the Company is so advised can be sold in (or during the time
               of) such

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               offering: first, such securities proposed by the stockholder
               exercising the demand registration right to be sold for its
               account; second, such securities requested to be included
               pursuant to incidental registration rights in such registration
               by the holder or holders, as the case may be, including the
               Holders, pro rata on the basis of the number of such securities
               so proposed to be sold by all such security holders and so
               requested to be included; and third, such securities proposed by
               the Company to be sold for its own account

                    (ii)  If (A) a registration pursuant to this Section 2.2
               involves an underwritten offering of the securities so being
               registered for sale for the account of the Company, to be
               distributed (on a firm commitment basis) by or through one or
               more underwriters of recognized standing under underwriting terms
               appropriate for such a transaction, and (B) the managing
               underwriter of such underwritten offering shall inform the
               Company and the Holders in writing of its belief that the number
               of securities requested to be included in such registration
               exceeds the number that can be sold in (or during the time of)
               such offering, then the Company will include in such
               registration, to the extent of the number that the Company is so
               advised can be sold in (or during the time of) such offering,
               securities proposed by the Company to be sold for its own
               account, and, such securities requested to be included pursuant
               to incidental registration rights in such registration by the
               holder or holders, as the case may be, including the Holders, pro
               rata on the basis of the number of such securities so proposed to
               be sold by all such security holders and so requested to be
               included.

               2.3  Registration Procedures. If and whenever the Company is
                    -----------------------
          required to use its best efforts to effect the registration of any
          Registrable Securities under the Securities Act as provided in
          Sections 2.1 and 2.2, the Company will as expeditiously as possible:

                    (a)  prepare and (as soon thereafter as possible or in any
               event no later than sixty (60) days after the end of the period
               within which requests for registration may be given to the
               Company) file with the Commission the requisite registration
               statement to effect such registration and thereafter use its best
               efforts to cause such registration statement to become effective,
               provided that the Company may discontinue any registration of its
               securities that are not Registrable Securities (and, under the
               circumstances specified in Section 2.2(a), its securities that
               are Registrable Securities) at any time prior to the effective
               date of the registration statement relating thereto;

                    (b)  prepare and file with the Commission such amendments
               and supplements to such registration statement and the prospectus
               used in connection therewith as may be necessary to keep such
               registration statement effective and to comply with the
               provisions of the Securities Act with respect to the disposition
               of all

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               securities covered by such registration statement until such time
               as all of such securities have been disposed of in accordance
               with the intended methods of disposition by the seller or sellers
               thereof set forth in such registration statement;

                    (c)  furnish to the Holders such number of conformed copies
               of such registration statement and of each such amendment and
               supplement thereto (in each case including all exhibits), such
               number of copies of the prospectus contained in such registration
               statement (including each preliminary prospectus and any summary
               prospectus) and any other prospectus filed under Rule 424 under
               the Securities Act, in conformity with the requirements of the
               Securities Act, and such other documents, as the Holders may
               reasonably request;

                    (d)  use its best efforts to register or qualify all
               Registrable Securities and other securities covered by such
               registration statement under such other securities or blue sky
               laws of such jurisdictions as the Holders shall reasonably
               request, to keep such registration or qualification in effect for
               so long as such registration statement remains in effect, and
               take any other action that may be reasonably necessary or
               advisable to enable the Holders to consummate the disposition in
               such jurisdictions of the securities owned by it, except that the
               Company shall not for any such purpose be required to qualify
               generally to do business as a foreign corporation in any
               jurisdiction wherein it would not but for the requirements of
               this subdivision (d) be obligated to be so qualified or to
               consent to general service of process in any such jurisdiction;

                    (e)  use its best efforts to cause all Registrable
               Securities covered by such registration statement to be
               registered with or approved by such other governmental agencies
               or authorities as may be necessary to enable the seller or
               sellers thereof to consummate the disposition of such Registrable
               Securities;

                    (f)  furnish to the Holders a signed counterpart, addressed
               to such seller (and underwriters, if any) of:

                         (i)   an opinion of counsel for the Company, dated the
                    effective date of such registration statement (and, if such
                    registration includes an underwritten public offering, dated
                    the date of the closing under the underwriting agreement),
                    reasonably satisfactory in form and substance to such
                    seller, and

                         (ii)  a "comfort" letter, dated the effective date of
                    such registration statement (and, if such registration
                    includes an underwritten public offering, dated the date of
                    the closing under the underwriting agreement), signed by the
                    independent public accountants who have certified the
                    Company's financial statements included in such registration
                    statement,

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               covering substantially the same matters with respect to such
               registration statement (and the prospectus included therein) and,
               in the case of the accountants' letter, with respect to events
               subsequent to the date of such financial statements, as are
               customarily covered in opinions of issuer's counsel and in
               accountants' letters delivered to the underwriters in
               underwritten public offerings of securities and, in the case of
               the accountants' letter, such other financial matters, and, in
               the case of the legal opinion, such other legal matters, as such
               seller may reasonably request;

                    (g)  notify the Holders, at any time when a prospectus
               relating thereto is required to be delivered under the Securities
               Act, upon discovery that, or upon the happening of any event as a
               result of which, the prospectus included in such registration
               statement, as then in effect, includes an untrue statement of a
               material fact or omits to state any material fact required to be
               stated therein or necessary to make the statements therein not
               misleading in the light of the circumstances under which they
               were made, and at the request of the Holders promptly prepare and
               furnish to the Holders a reasonable number of copies of a
               supplement to or an amendment of such prospectus as may be
               necessary so that, as thereafter delivered to the purchasers of
               such securities, such prospectus shall not include an untrue
               statement of a material fact or omit to state a material fact
               required to be stated therein or necessary to make the statements
               therein not misleading in the light of the circumstances under
               which they were made;

                    (h)  otherwise use its best efforts to comply with all
               applicable rules and regulations of the Commission, and make
               available to its security holders, as soon as reasonably
               practicable, an earnings statement covering the period of at
               least twelve months, but not more than eighteen months, beginning
               with the first full calendar month after the effective date of
               such registration statement, which earnings statement shall
               satisfy the provisions of Section 11(a) of the Securities Act,
               and will furnish to the Holders at least five (5) business days
               prior to the filing thereof a copy of any amendment or supplement
               to such registration statement or prospectus and shall not file
               any thereof to which the Holders shall have reasonably objected
               on the grounds that such amendment or supplement does not comply
               in all material respects with the requirements of the Securities
               Act or of the rules or regulations thereunder;

                    (i)  provide and cause to be maintained a transfer agent and
               registrar for all Registrable Securities covered by such
               registration statement from and after a date not later than the
               effective date of such registration statement;

                    (j)  use its best efforts to list all Registrable Securities
               covered by such registration statement on any securities exchange
               or automated quotation system on which any of the Registrable
               Securities is then listed; and

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                    (k)  enter into such agreements and take such other actions
               as the Holders shall reasonably request in order to expedite or
               facilitate the disposition of such Registrable Securities.

The Company may require the Holders participating in a registration to furnish
the Company such information regarding such Holder and the distribution of such
securities as the Company may from time to time reasonably request in writing.

     Each Holder agrees that upon receipt of any notice from the Company of the
happening of any event of the kind described in subdivision (g) of this Section
2.3, the Holder will forthwith discontinue its disposition of Registrable
Securities pursuant to the registration statement relating to such Registrable
Securities until the Holder's receipt of the copies of the supplemented or
amended prospectus contemplated by subdivision (g) of this Section 2.3 and, if
so directed by the Company, will deliver to the Company (at the Company's
expense) all copies, other than permanent file copies, then in the Holder's
possession of the prospectus relating to such Registrable Securities current at
the time of receipt of such notice.

               2.4  Underwritten Offerings.
                    ----------------------

                    (a)  Requested Underwritten Offerings. If requested by the
                         --------------------------------
               underwriters for any underwritten offering by a Holder pursuant
               to a Demand Registration requested under Section 2.1, the Company
               will enter into an underwriting agreement with such underwriters
               for such offering, such agreement to be reasonably satisfactory
               in substance and form to the Holder and the underwriters and to
               contain such representations and warranties by the Company and
               such other terms as are generally prevailing in agreements of
               this type, including, without limitation, indemnities to the
               effect and to the extent provided in Section 2.6. The Holder
               shall be a party to such underwriting agreement and may, at its
               option, require that any or all of the representations and
               warranties by, and the other agreements on the part of, the
               Company to and for the benefit of such underwriters shall also be
               made to and for the benefit of the Holder and that any or all of
               the conditions precedent to the obligations of such underwriters
               under such underwriting agreement be conditions precedent to the
               obligations of the Holders. The Holders shall not be required to
               make any representations or warranties to or agreements with the
               Company or the underwriters other than representations,
               warranties or agreements regarding the Holders, its Registrable
               Securities and its intended method of distribution and any other
               representation required by law.

                    (b)  Incidental Underwritten Offerings. If the Company at
                         ---------------------------------
               any time proposes to register any of its securities under the
               Securities Act whether or not for sale for its own account as
               contemplated by Section 2.2 and such securities are to be
               distributed by or through one or more underwriters, the Company
               will, if requested by the Holders as provided in Section 2.2 and
               subject to the provisions of Section

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               2.2(b), arrange for such underwriters to include all the
               Registrable Securities to be offered and sold by the Holders
               among the securities to be distributed by such underwriters. The
               participating Holders shall be a party to the underwriting
               agreement between the Company and such underwriters and may, at
               its option, require that any or all of the representations and
               warranties by, and the other agreements on the part of, the
               Company to and for the benefit of such underwriters shall also be
               made to and for the benefit of the participating Holders and that
               any or all of the conditions precedent to the obligations of such
               underwriters under such underwriting agreement be conditions
               precedent to the obligations of the Holders. The Holders shall
               not be required to make any representations or warranties to or
               agreements with the Company or the underwriters other than
               representations, warranties or agreements regarding the Holders,
               its Registrable Securities and its intended method of
               distribution and any other representation required by law.

               2.5  Preparation; Reasonable Investigation. In connection with
                    -------------------------------------
          the preparation and filing of each registration statement under the
          Securities Act pursuant to this Agreement, the Company will give the
          Holders, and their counsel and accountants, the opportunity to
          participate in the preparation of such registration statement, each
          prospectus included therein or filed with the Commission, and each
          amendment thereof or supplement thereto, and will give it such access
          to its books and records and such opportunities to discuss the
          business of the Company with its officers and the independent public
          accountants who have certified its financial statements as shall be
          necessary, in the opinion of the Holders' counsel, to conduct a
          reasonable investigation within the meaning of the Securities Act.

               2.6  Indemnification.
                    ---------------

                    (a)  Indemnification by the Company. In the event of any
                         ------------------------------
               registration of any securities of the Company under the
               Securities Act, the Company will, and hereby does, in the case of
               any registration statement filed pursuant to Section 2.1 or 2.2,
               indemnify and hold harmless the Holders, its directors and
               officers, each other Person who participates in the offering or
               sale of such securities and each other Person, if any, who
               controls any Holder within the meaning of the Securities Act
               against any losses, claims, damages or liabilities, joint or
               several, to which the Holders or any such director or officer or
               controlling person may become subject under the Securities Act or
               otherwise, insofar as such losses, claims, damages or liabilities
               (or actions or proceedings, whether commenced or threatened, in
               respect thereof) arise out of or are based upon any untrue
               statement or alleged untrue statement of any material fact
               contained in any registration statement under which such
               securities were registered under the Securities Act, any
               preliminary prospectus, final prospectus or summary prospectus
               contained therein, or any amendment or supplement thereto, or any
               omission or alleged omission to state therein a material fact
               required to be stated therein or necessary to make the statements
               therein not misleading, and the Company will reimburse the
               Holders and each such director,

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               officer, and controlling person for any legal or any other
               expenses reasonably incurred by them in connection with
               investigating or defending any such loss, claim, liability,
               action or proceeding; provided that the Company shall not be
                                     --------
               liable in any such case to the extent that any such loss, claim,
               damage, liability (or action or proceeding in respect thereof) or
               expense arises out of or is based upon an untrue statement or
               alleged untrue statement or omission or alleged omission made in
               such registration statement, any such preliminary prospectus,
               final prospectus, summary prospectus, amendment or supplement in
               reliance upon and in conformity with written information
               furnished to the Company through an instrument duly executed by a
               Holder specifically stating that it is for use in the preparation
               thereof and, provided further that the Company shall not be
                            --------
               liable to any Person who participates as an underwriter, in the
               offering or sale of Registrable Securities or any other Person,
               if any, who controls such underwriter within the meaning of the
               Securities Act, in any such case to the extent that any such
               loss, claim, damage, liability (or action or proceeding in
               respect thereof) or expense arises out of such Person's failure
               to send or give a copy of the final prospectus, as the same may
               be then supplemented or amended, to the Person asserting an
               untrue statement or alleged untrue statement or omission or
               alleged omission at or prior to the written confirmation of the
               sale of Registrable Securities to such Person if such statement
               or omission was corrected in such final prospectus. Such
               indemnity shall remain in full force and effect regardless of any
               investigation made by or on behalf of any Holder or any such
               director, officer, underwriter or controlling person and shall
               survive the transfer of such securities by such Holder.

                    (b)  Indemnification by the Holders. The Company may
                         ------------------------------
               require, as a condition to including any Registrable Securities
               in any registration statement filed pursuant to Section 2.6, that
               the Company shall have received an undertaking satisfactory to it
               from each participating Holder, to indemnify and hold harmless
               (in the same manner and to the same extent as set forth in
               subdivision (a) of this Section 2.7) the Company, each director
               of the Company, each officer of the Company and each other
               Person, if any, who controls the Company within the meaning of
               the Securities Act, with respect to any statement or alleged
               statement in or omission or alleged omission from such
               registration statement, any preliminary prospectus, final
               prospectus or summary prospectus contained therein, or any
               amendment or supplement thereto, if such statement or alleged
               statement or omission or alleged omission was made in reliance
               upon and in conformity with written information furnished to the
               Company through an instrument duly executed by the Holder
               specifically stating that it is for use in the preparation of
               such registration statement, preliminary prospectus, final
               prospectus, summary prospectus, amendment or supplement. Such
               indemnity shall remain in full force and effect, regardless of
               any investigation made by or on behalf of the Company or any such
               director, officer or controlling Person and shall survive the
               transfer of such securities by any Holder.

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                    (c)  Notices of Claims, etc. Promptly after receipt by an
                         ----------------------
               indemnified party of notice of the commencement of any action or
               proceeding involving a claim referred to in the preceding
               subdivisions of this Section 2.6, such indemnified party will, if
               a claim in respect thereof is to be made against an indemnifying
               party, give written notice to the latter of the commencement of
               such action, provided that the failure of any indemnified party
               to give notice as provided herein shall not relieve the
               indemnifying party of its obligations under the preceding
               subdivisions of this Section 2.7, except to the extent that the
               indemnifying party is actually prejudiced by such failure to give
               notice. In case any such action is brought against an indemnified
               party, unless in such indemnified party's reasonable judgment a
               conflict of interest between such indemnified and indemnifying
               parties may exist in respect of such claim, the indemnifying
               party shall be entitled to participate in and to assume the
               defense thereof, jointly with any other indemnifying party
               similarly notified to the extent that it may wish, with counsel
               reasonably satisfactory to such indemnified party, and after
               notice from the indemnifying party to such indemnified party of
               its election so to assume the defense thereof, the indemnifying
               party shall not be liable to such indemnified party for any legal
               or other expenses subsequently incurred by the latter in
               connection with the defense thereof other than reasonable costs
               of investigation. No indemnifying party shall, without the
               consent of the indemnified party, consent to entry of any
               judgment or enter into any settlement that does not include as an
               unconditional term thereof the giving by the claimant or
               plaintiff to such indemnified party of a release from all
               liability in respect to such claim or litigation.

                    (d)  Other Indemnification. Indemnification similar to that
                         ---------------------
               specified in the preceding subdivisions of this Section 2.6 (with
               appropriate modifications) shall be given by the Company and the
               Holders with respect to any required registration or other
               qualification of securities under any Federal or state law or
               regulation of any governmental authority other than the
               Securities Act.

                    (e)  Indemnification Payments. The indemnification required
                         ------------------------
               by this Section 2.6 shall be made by periodic payments of the
               amount thereof during the course of the investigation or defense,
               as and when bills are received or expense, loss, damage or
               liability is incurred.

               2.7  Adjustments Affecting Registrable Securities. The Company
                    --------------------------------------------
          will not effect or permit to occur any combination or subdivision of
          Registrable Securities which would adversely affect the ability of the
          Holders of Registrable Securities to include such Registrable
          Securities in any registration of its securities contemplated by this
          Section 2 or the marketability of such Registrable Securities under
          any such registration.

          3.   Definitions. As used herein, unless the context otherwise
               -----------
requires, the following terms have the following respective meanings:

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               Commission: The Securities and Exchange Commission or any other
               ----------
               federal agency at the time administering the Securities Act.

               Company: As defined in the introductory paragraph of this
               -------
               Agreement. For purposes of this Agreement, all references to the
               Company shall be deemed to include any successor entity or
               transferee.

               Exchange Act: The Securities Exchange Act of 1934, or any similar
               ------------
               federal statute, and the rules and regulations of the Commission
               thereunder, all as the same shall be in effect from time to time.
               Reference to a particular section of the Exchange Act shall
               include a reference to the comparable section, if any, of any
               other similar federal statute.

               Fair Market Value: With respect to a calculation of the value of
               -----------------
               Registrable Securities, the product of the number of such
               Registrable Securities and the average of the closing price of
               the Common Stock on all domestic securities exchanges on which
               such security may on the date of calculation be listed, or, if
               there have been no sales on any such exchange on the day of
               request, the average of the highest bid and lowest asked prices
               on all such exchanges at the end of such day, or, if on such 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.

               Holder: Union Oil and any Person to which Union Oil transfers
               ------
               Registrable Securities.

               Initial Registration Date: The date that is 120 days after the
               -------------------------
               date on which the Effective Time of the Merger (each as defined
               in the Merger Agreement) occurs.

               Person: A corporation, an association, a partnership, a business,
               ------
               an individual, a governmental or political subdivision thereof or
               a governmental agency.

               Registrable Securities: Any of the Company's Common Stock issued
               ----------------------
               to Union Oil pursuant to or in connection with the Merger
               Agreement or the Non-Dilution Agreement, and any securities
               issued to issuable with respect to any such Common Stock by way
               of distribution or in connection with any reorganization,
               recapitalization, merger, consolidation or otherwise. As to any
               particular Registrable Securities, once issued such securities
               shall cease to be Registrable Securities when (a) a registration
               statement with respect to the sale of such securities shall have

                                      -11-
<PAGE>

               become effective under the Securities Act and such securities
               shall have been disposed of in accordance with such registration
               statement, (b) they shall have been distributed to the public
               pursuant to Rule 144 or Rule 144A (or any successor provision)
               under the Securities Act, (c) with respect to any Holder, the
               Registrable Securities held by such Holder represent less than 2%
               of the then issued and outstanding shares of Common Stock, such
               Holder is not an "affiliate" of the Company, and such shares may
               be sold freely by such Holder without restriction under Rule
               144(k) (or any successor provision) without restriction, or (d)
               they shall have ceased to be outstanding.

               Registration Expenses: All expenses incident to the Company's
               ---------------------
               performance of or compliance with Section 2.1 or 2.2, including,
               without limitation, all salaries of Company personnel or general
               overhead expenses of the Company, auditing fees, premiums or
               other expenses relating to liability insurance required by
               underwriters of the Company, or other expenses for the
               preparation of financial statements, all registration, filing and
               National Association of Securities Dealers fees, all fees and
               expenses of complying with securities or blue sky laws, all word
               processing, duplicating and printing expenses, messenger and
               delivery expenses, the fees and disbursements of counsel for the
               Company and of its independent public accountants, including the
               expenses of any special audits or "cold comfort" letters required
               by or incident to such performance and compliance, the fees and
               disbursements of not more than one special counsel to the holders
               of such Registrable Securities not to exceed $50,000 per
               registration, premiums and other costs of policies of insurance
               against liabilities arising out of the public offering of the
               Registrable Securities being registered and any fees and
               disbursements of underwriters customarily paid by issuers or
               sellers of securities, but excluding Selling Expenses.

               Securities Act: The Securities Act of 1933, or any similar
               --------------
               federal statute, and the rules and regulations of the Commission
               thereunder, all as of the same shall be in effect from time to
               time. References to a particular section of the Securities Act
               shall include a reference to the comparable section, if any, of
               any other similar federal statute.

               Selling Expenses: underwriting discounts and commissions and
               ----------------
               stock transfer taxes relating to securities registered by the
               Holders.

               Union Oil: As defined in the introductory paragraph of this
               ---------
               Agreement.

          4.   Rule 144 and Rule 144A: The Company will file the reports
               ----------------------
required to be filed by it under the Securities Act and the Exchange Act and the
rules and regulations adopted by the Commission thereunder (or, if the Company
is not required to file such reports, will, upon the request of any holder of
Registrable Securities, make publicly available other information) and will take
such further action as the Holders may reasonably request, all to the extent
required from time

                                      -12-
<PAGE>

to time to enable the Holders to sell Registrable Securities without
registration under the Securities Act within the limitation of the exemptions
provided by (a) Rule 144 under the Securities Act, as such rule may be amended
from time to time or (b) any similar rule or regulation hereafter adopted by the
Commission. Upon the request of any Holder, the Company will deliver to such
holder a written statement as to whether it has complied with such requirements.
After any sale of Registrable Securities pursuant to this Section 4, the Company
will, to the extent allowed by law, cause any restrictive legends to be removed
and any transfer restrictions to be rescinded with respect to such Registrable
Securities. In order to permit the Holders to sell the same, if it so desires,
pursuant to Rule 144A promulgated by the Commission (or any successor to such
rule), the Company will comply with all rules and regulations of the Commission
applicable in connection with use of Rule 144A (or any successor thereto).
Prospective transferees of Registrable Securities that are Qualified
Institutional Buyers (as defined in Rule 144A) which would be purchasing such
Registrable Securities in reliance upon Rule 144A may request from the Company
information regarding the business, operations and assets of the Company. Within
five business days of any such request, the Company shall deliver to any such
prospective transferee copies of annual audited and quarterly unaudited
financial statements of the Company and such other information as may be
required to be supplied by the Company for it to comply with Rule 144A.

          5.   Amendments and Waivers. This Agreement may be amended and the
               ----------------------
Company may take any action herein prohibited or omit to perform any act herein
required to be performed by it, only if the Company shall have obtained the
written consent to such amendment, action or omission to act, of the Holders of
a majority of the Registrable Securities. The Holders of any Registrable
Securities at the time or thereafter outstanding shall be bound by any consent
authorized by this Section 5, whether or not such Registrable Securities shall
have been marked to indicate such consent.

          6.   Nominees for Beneficial Holders. In the event that any
               -------------------------------
Registrable Securities are held by a nominee for the beneficial owner thereof,
the beneficial owner thereof may, at its election, be treated as the holder of
such Registrable Securities for purposes of this Agreement. If the beneficial
owner of any Registrable Securities so elects, the Company may require
assurances reasonably satisfactory to it of such owner's beneficial ownership of
such Registrable Securities.

          7.   Notices. All communications provided for hereunder shall be sent
               -------
by first-class mail and (a) if addressed to Union Oil, at One Sugar Creek Place,
14141 Southwest Freeway, Sugar Land, Texas 77478, Attention: Mr. Phil Ballard,
(b) if addressed to any other Holder, to the address specified by such Holder
upon its receipt of Registrable Securities, or (c) if addressed to the Company,
at 500 West Texas, Suite 200, Midland, Texas 79701 or at such other address, or
to the attention of such other officer, as the Company shall have furnished to
the Holders at the time outstanding; provided, however, that any such
communication to the Company may also, at the option of any Holder, be either
delivered to the Company at its address set forth above or to any officer of the
Company.

                                      -13-
<PAGE>

          8.   Assignment. This Agreement shall be binding upon and inure to the
               ----------
benefit of and be enforceable by the parties hereto and their respective
successors and assigns. In addition, and whether or not any express assignment
shall have been made, the provisions of this Agreement that are for the benefit
of any Holder shall also be for the benefit of and enforceable by any subsequent
holder of any Registrable Securities, subject to the provisions respecting the
minimum numbers or percentages of shares of Registrable Securities required in
order to be entitled to certain rights, or take certain actions, contained
herein.

          9.   Effectiveness; Termination. This Agreement shall become effective
               --------------------------
only upon consummation of the transactions contemplated by the Merger Agreement,
and shall terminate if the Merger Agreement is terminated in accordance with its
terms. After this Agreement becomes effective, this Agreement shall terminate
when no Registrable Securities remain outstanding.

          10.  Descriptive Headings. The descriptive headings of the several
               --------------------
sections and paragraphs of this Agreement are inserted for reference only and
shall not limit or otherwise affect the meaning hereof.

          11.  Specific Performance. The parties hereto recognize and agree that
               --------------------
money damages may be insufficient to compensate the Holders for breaches by the
Company of the terms hereof and, consequently, that the equitable remedy of
specific performance of the terms hereof will be available in the event of any
such breach.

          12.  Governing Law. THIS AGREEMENT SHALL BE CONSTRUED AND ENFORCED IN
               -------------
ACCORDANCE WITH, AND THE RIGHTS OF THE PARTIES SHALL BE GOVERNED BY, THE LAWS OF
THE STATE OF DELAWARE WITHOUT REGARD TO PRINCIPLES OF CONFLICTS OF LAWS.

          13.  Counterparts. This Agreement may be executed simultaneously in
               ------------
any number of counterparts, each of which shall be deemed an original, but all
such counterparts shall together constitute one and the same instrument.

                                      -14-
<PAGE>

     IN WITNESS WHEREOF, the parties have caused this Agreement to be executed
and delivered by their respective officers thereunto duly authorized as of the
date first above written.

                                        COMPANY:

                                        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

                                      -15-