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                                                                   EXHIBIT 10.12

                            CONVEYANCE OF ADJUSTABLE
                        TERM OVERRIDING ROYALTY INTEREST

         This Conveyance of Adjustable Term Overriding Royalty Interest (this
"Conveyance"), is by and between Brigham Oil & Gas, L.P., a Delaware limited
partnership (the "Grantor"), and ECT Merchant Investments Corp., a Delaware
corporation ("ECT Merchant") and Joint Energy Development Investments II
Limited Partnership, a Delaware limited partnership ("JEDI-II and, together
with ECT Merchant, the "Grantee").

                                   RECITALS:

         1. Brigham Exploration Company ("Brigham Parent") and Chase Bank of
Texas, National Association, as Trustee, have entered into an Indenture dated
as of August 20, 1998, as amended by a First Amendment to Indenture dated as of
March 26, 1999, and a Second Amendment to Indenture dated as of even date
herewith (as amended, the "Indenture"), pursuant to which Brigham Parent issued
certain Notes (as defined in the Indenture).

         2. In accordance with the terms of the Indenture, Brigham Parent
agreed to convey, and to cause its Subsidiaries to convey, to the Noteholders
(as defined in the Indenture), Term ORRI's (as defined below), as an absolute
assignment, and not as collateral, such conveyance to be made pursuant to one
or more Term ORRI Conveyances (as defined in the Indenture).

         3. This Conveyance constitutes one of the Term ORRI Conveyances
contemplated by the Indenture.

                                   ARTICLE I.

                                 DEFINED TERMS

         Section 1. Defined Terms and References. Reference is hereby made to
the Indenture for all purposes. Unless otherwise defined herein, all
capitalized terms used herein shall have the meaning attributed to such term in
the Indenture. In addition, as used in this Conveyance, the following
capitalized terms shall have the following meanings:

         "Accrued PIK Amount" means, as of any particular day, the aggregate
dollar amount of all accrued interest on the Notes, that has been paid in kind
(and not in cash) as of such day in accordance with the Indenture.

         "Applicable Percentage" shall mean, with respect to any Subject
Reserves, Grantor's actual net revenue interest (expressed as a percentage) in
the production of oil, gas and/or other minerals in, under and/or that may be
produced from any Subject Reserves, (including without limitation any
net revenue interest attributable to an overriding royalty interest or acquired
by virtue of a third party

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nonconsent election in any drilling, completion or other operation), subject to
the agreements and other matters set out in Exhibit B hereto, and the rights
afforded third parties (which are not affiliates or Subsidiaries of Brigham
Parent) thereunder.

         "Brigham Parent" is defined in the Recitals.

         "Effective Date" shall mean 7:00 a.m. local time on February 17, 2000.

         "Hydrocarbon Interests" means all rights, titles and estates of
Grantor in and to the oil and gas leases, oil, gas and mineral leases, or other
liquid or gaseous hydrocarbon leases, mineral fee interests, overriding royalty
and royalty interests, net profit interests and production payment interests
identified on Exhibit A attached hereto or otherwise owned by Grantor,
including any reserved or residual interest of whatever nature.

         "Indenture" is defined in the Recitals.

         "Marketing Terms and Conditions" shall have the meaning assigned to it
in Section 4.1.

         "Non-Proved Reserves" means any category of oil and gas reserves other
than Proved Reserves.

         "ORRI Hydrocarbons" shall have the meaning assigned to it in Section
4.1.

         "ORRI Percentage" shall be defined as follows:

                  (a)      Until the occurrence of the Recalculation Event, the
                           ORRI Percentage shall be four percent (4%).

                  (b)      Under circumstances more particularly described in
                           Section 2.3, effective from and after the occurrence
                           of the Recalculation Event, the ORRI Percentage
                           shall be adjusted downward to three percent (3%).

         "ORRI Term" means the period from the Effective Date to a date that is
the earliest to occur of (a) the date on which there is a payment in full of
the Accrued PIK Amount and all other Obligations under the Indenture or (b) the
date on or after November 21, 2000, on which there is a payment in full of the
Accrued PIK Amount.

         "Overproduced Position" shall mean a position that arises as a result
of a party (and/or its predecessors in title) taking more production from a
property than its ownership interest in such property would, but for a gas
balancing or similar agreement, entitle it to take.

         "Permissible Charges" shall mean (i) all production, severance, ad
valorem or similar taxes assessed against or measured by production and
severance of ORRI Hydrocarbons or the value and (ii) with respect to each well
not operated by the Grantor or any Affiliate thereof, the Term ORRI's allocable
share of any reasonable post production expenses, costs, charges, and fees
incurred in

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transporting, processing, gathering, treating, separating, dehydrating and
marketing the Subject Reserves allocable to such well.

         "PDNP Reserves" shall mean proved, developed, non-producing oil and
gas reserves, as determined in conformity with the guidelines in effect from
time to time as promulgated by the Society of Petroleum Engineers or its
successor association.

         "PDP Reserves" shall mean proved, developed, producing oil and gas
reserves, as determined in conformity with the guidelines in effect from time
to time as promulgated by the Society of Petroleum Engineers or its successor
association.

         "Proved Reserves" means PDP Reserves, PDNP Reserves and PUD Reserves.

         "PUD Reserves" shall mean proved, undeveloped oil and gas reserves, as
determined in conformity with the guidelines in effect from time to time as
promulgated by the Society of Petroleum Engineers or its successor association.

         "Recalculation Event" shall have the meaning assigned to it in Section
2.3.

         "Subject Reserves" means (a) any PDP Reserves and PDNP Reserves
identified by Grantor or Brigham Parent (or any Subsidiary thereof) as of the
date hereof as being produced or producible in paying quantities out of wells
now located on the Hydrocarbon Interests as of the date hereof including those
listed on the attached Exhibit A or (b) any PDP Reserves and PDNP Reserves
identified by Grantor or Brigham Parent (or any Subsidiary thereof) following
the date hereof and on or before December 31, 2000, as being produced or
producible in paying quantities out of any well spudded or acquired by Grantor
or Brigham Parent (or any Subsidiary thereof) on or before December 31, 2000,
(c) in the event only that Brigham Parent's Consolidated Interest Coverage
Ratio calculated as of September 30, 2000 (in accordance with the Indenture),
is less than 1.1 to 1.0, any PDP Reserves and PDNP Reserves identified by
Grantor or Brigham Parent (or any Subsidiary thereof) following December 31,
2000 and on or before December 31, 2001, as being produced or producible in
paying quantities out of any well spudded or acquired after December 31, 2000
but on or before December 31, 2001, and (d) in the cases of (a), (b) and (c),
above, a sufficient interest in the Hydrocarbon Interests to afford Grantor or
the Brigham Parent as applicable the right, prior to giving effect hereto or to
any Term ORRI Conveyance, to secure the maximum authorized share of production
that is allocable to PDP Reserves and, when applicable, PDNP Reserves found in
each Subject Well. Under no circumstances, and notwithstanding any provision
hereof to the contrary, shall the term "Subject Reserves" include (1) any PUD
Reserves or any Non-Proved Reserves or (2) any PDP Reserves and/or PDNP
Reserves beneficially or legally owned (whether in a vested or contingent
manner) by a third party (which is not an Affiliate of the Brigham Parent)
under or pursuant to any agreement or other matter disclosed in Exhibit B
hereto, or (3) any PDP Reserves or PDNP Reserves identified by Brigham Parent
or Grantor as a result of the deepening or sidetracking of a Subject Well after
December 31, 2000 (in the event item (c) of the definition of Subject Reserves
is not triggered) and after December 31, 2001, (in the event item (c) of the
definition of Subject Reserves is triggered). Under no circumstances shall the
term "Subject Reserves" include reserves that are not either producing or
capable of producing (or being made in

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the case of PDNP Reserves to produce or capable of producing) in "paying
quantities," as such term is defined under the Texas common law (regardless of
the jurisdiction in which the subject Oil and Gas Properties are located).

         "Term ORRI" shall have the meaning assigned to it in Section 2.1.

         "Underproduced Position" shall mean a position that arises as a result
of a party (and/or its predecessors in title) taking less production from a
property than its ownership interest in such property would, but for a gas
balancing or similar agreement, entitle it to take.

                                  ARTICLE II.

                              GRANTING PROVISIONS

         Section  2.1      Granting Clause. For a good and valuable
                           consideration, the receipt and sufficiency of which
                           is hereby acknowledged, Grantor does hereby GRANT,
                           BARGAIN, SELL, TRANSFER, ASSIGN, CONVEY, WARRANT and
                           DELIVER, for and during the ORRI Term only, to each
                           Grantee according to their Participations, an
                           overriding royalty interest (collectively, the "Term
                           ORRI") equal to the ORRI Percentage of the
                           Applicable Percentage of each of the Subject
                           Reserves.

         Section  2.2      Adjustment of the Term ORRI; Further Assurances.
                           Upon the occurrence of the Recalculation Event,
                           Grantor and Grantee agree, at Grantor's expense, to
                           execute (or cause to be executed) any such
                           instruments as may be necessary or appropriate in
                           order to evidence of record in the appropriate
                           filing jurisdictions that the ORRI Percentage has
                           been adjusted.

         Section 2.3       Adjustment of ORRI Percentage. The ORRI Percentage
                           shall be automatically decreased from four (4%) to
                           three (3%) on the first to occur of the following
                           (the "Recalculation Event"):

               (a)  Brigham Parent sells common or preferred equity securities
                    of Brigham Parent resulting in net proceeds received by
                    Brigham Parent (after deducting all costs and expenses
                    incurred in such sale) of at least $10,000,000;

               (b)  Grantor and/or Brigham Parent acquires, in exchange for
                    common or preferred equity securities of Brigham Parent,
                    Oil and Gas Properties containing PDP Reserves and/or PDNP
                    Reserves with a net present value, discounted at 10% per
                    annum, of at least $10,000,000 (in Grantees' reasonable
                    judgement); or

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               (c)  through a combination of (a) and (b) above, Brigham Parent
                    and/or Grantor receives net proceeds and newly acquired Oil
                    and Gas Properties, which (as valued above), total at least
                    $10,000,000 in net present value;

provided that, on or before August 31, 2000, the Recalculation Event shall have
closed and (a) Brigham Parent shall have received the $10,000,000 of cash
and/or properties as provided above, and (b) either the Chief Financial Officer
or President of Brigham Parent shall have delivered a certificate to Grantor to
that effect.

         Section 2.4       Term. The Term ORRI shall automatically terminate at
                           the end of the ORRI Term. Upon such event, Grantee
                           shall execute such instruments as Grantor shall
                           reasonably request to evidence such termination,
                           and/or reassigning such Term ORRI, free and clear of
                           all liens, encumbrances and other burdens created
                           by, through or under Grantee.

         Section 2.5       Permissible Charges. The Term ORRI shall be free of
                           any expense for exploration, drilling, development,
                           operating, marketing and other costs incident to the
                           production and sale of oil and gas other than
                           Permissible Charges. Notwithstanding any provision
                           hereof to the contrary, the Term ORRI shall not
                           apply to (a) any oil, gas and/or other minerals as
                           are (i) unavoidably lost in operations on or
                           relating to the Subject Reserves or in marketing of
                           oil, gas and/or other minerals from or attributable
                           to the Subject Reserves, or (ii) consumed or
                           utilized (e.g., as part of a re-injection program)
                           in prudent operations conducted on the Subject
                           Reserves with a reasonable expectation of increasing
                           ultimate recovery of oil, gas and/or other minerals
                           from the Subject Reserves; provided that, to the
                           extent any oil, gas and/or other minerals utilized
                           in operations conducted on the Subject Reserves are
                           subsequently re-produced, the Term ORRI shall apply
                           to such re-produced volumes.

         Section 2.6       Pooling Issues. Prior to the date hereof, certain of
                           the Hydrocarbon Interests may have been pooled or
                           unitized for the production of oil, gas and/or
                           minerals and without the joinder of Grantee with
                           respect to the Term ORRI, Grantor shall, from and
                           after the date hereof, have the right and power to
                           unitize, communitize or pool any portion or portions
                           of the Term ORRI. If pursuant to any law, rule,
                           regulation or order of any governmental body or
                           official, any portion of the Hydrocarbon Interests
                           is automatically or involuntarily pooled,
                           communitized or unitized in any manner, the Term
                           ORRI, insofar as it affects such Hydrocarbon
                           Interests, shall also be pooled, communitized and
                           unitized. With respect to each existing pool or
                           unit, and with respect to each pool or unit in which
                           the Term ORRI is hereafter included in accordance
                           with the

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                           foregoing, the Term ORRI in each portion of the
                           Hydrocarbon Interests included in such pool or unit
                           shall apply to the portion of production from such
                           pool or unit which is attributable to such portion
                           of the Hydrocarbon Interests under and by virtue of
                           the applicable pooling, communitization and
                           unitization agreements, designations and/or orders.
                           Nothing in this Section 2.6 shall be construed to
                           afford Grantee rights in PDP Reserves or PDNP
                           Reserves identified in wells that are not now
                           located on the Hydrocarbon Interests.

         Section 2.7       Renewals and Extensions. The Term ORRI shall apply
                           to all renewals and extensions of any oil, gas
                           and/or mineral lease, deed or any other interest
                           included, in part or in whole, in the Subject
                           Reserves.

         Section 2.8       Gas Imbalances. It is hereby recognized that certain
                           of the Subject Reserves are or may be subject to
                           Underproduced Positions. It is hereby further
                           recognized and stipulated that the Term ORRI does
                           and shall apply to the volumes of oil, gas or other
                           hydrocarbons taken by Grantor in reducing or making
                           up an Underproduced Position (which volumes and any
                           make-up rights relating thereto shall for all
                           purposes hereof be included within the Subject
                           Reserves), whether the subject Underproduced
                           Position arose before, or arises after, the
                           Effective Date hereof.

         Section 2.9       Fee, Royalty and Similar Interests. It is hereby
                           recognized that certain of the Hydrocarbon Interests
                           may constitute or include (a) unleased mineral fee
                           interests and unleased mineral rights or servitudes
                           (the "Unleased Mineral Interests"), (b) mineral fee
                           interests and mineral rights or servitudes that are
                           owned by Grantor but subject to an existing oil, gas
                           and/or mineral lease or similar agreement in favor
                           of a third party or parties, which oil, gas and/or
                           mineral lease or similar agreement entitles Grantor
                           to be paid a royalty, either in kind or in money
                           (the "Leased Mineral Interests"), and (c) perpetual
                           or term royalty or overriding royalty interests
                           carved out of and burdening a mineral fee, right,
                           servitude or lease owned by a third party or third
                           parties (the "Royalty Interests") (the Unleased
                           Mineral Interests, the Leased Mineral Interests and
                           the Royalty Interests are herein collectively called
                           the "Non-Working Interest Properties"). It is hereby
                           recognized and stipulated that the Term ORRI is
                           payable out of and shall burden the Non-Working
                           Interest Properties, to the extent they constitute
                           Subject Reserves, just as it is payable out of and
                           burdens the balance of the Hydrocarbon Interests. By
                           way of example (and not by way of limitation):

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                  (i)      If Grantor owns an Unleased Mineral Interest that
                           constitutes a 50% undivided mineral fee interest in
                           a particular tract of land, the Term ORRI payable
                           relative to such tract of land would, until the
                           occurrence of the Recalculation Event, be equal to
                           and measured by an undivided 4% of 50% of the oil,
                           gas and/or other minerals in, under and/or that may
                           be produced from the subject tract of land;

                  (ii)     If Grantor owns a Leased Mineral Interest that
                           constitutes an undivided 50% mineral fee interest in
                           a particular tract of land that is subject to a
                           lease entitling Grantor to a 20% royalty interest,
                           proportionately reduced, the Term ORRI payable
                           relative to such tract of land would, until the
                           occurrence of the Recalculation Event, be equal to
                           and measured by an undivided 4% of 10% of the oil,
                           gas and/or other minerals in, under and/or that may
                           be produced from the subject tract of land; and

                  (iii)    If Grantor owns a Royalty Interest that constitutes
                           a 1/32nd nonparticipating royalty interest in a
                           particular tract of land, the Term ORRI payable
                           relative to such tract of land would, until the
                           occurrence of the Recalculation Event, be equal to
                           and measured by an undivided 4% of 1/32nd of the
                           oil, gas and/or other minerals in, under and/or that
                           may be produced from the subject tract of land.

         Section 2.10      Habendum Clause and Subrogation. TO HAVE AND TO HOLD
                           the Term ORRI unto Grantee, its successors and
                           assigns, during the ORRI Term. This Conveyance is
                           made with full substitution and subrogation of
                           Grantee in and to all covenants and warranties by
                           others heretofore given or made.

                                  ARTICLE III.

                           ASSIGNMENTS AND TRANSFERS

         Section 3.1       Assignment and Transfer by Grantee. Subject to
                           Section 7.12(e) of the Indenture, nothing herein
                           contained shall in any way limit or restrict the
                           right of Grantee to sell, convey, assign, mortgage
                           or otherwise dispose of the Term ORRI (including its
                           rights, titles, interests, estates, remedies, powers
                           and privileges appurtenant or incident to the Term
                           ORRI under this Conveyance), in whole or in part.

         Section 3.2       Assignment and Transfer by Grantor.

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         (A) In the event any of the Subject Reserves are sold, farmed out or
otherwise transferred by the Grantor in a bona fide transaction with a
third-party that is not an Affiliate of Grantor, the Term ORRI conveyed herein
in the Subject Reserves so sold, farmed out or otherwise transferred shall
automatically terminate and merge into the interest so sold, farmed-out or
otherwise transferred upon the final closing of such transaction; provided that
the Grantor pays or otherwise affords to Grantee, in accordance with their
Participations, (a) the ORRI Percentage of all net cash proceeds or other
consideration received (or credited) to Grantor as a result of such
transaction, to the extent only such net cash proceeds are allocable to the
Subject Reserves out of which the subject Term ORRI is carved and (b) the ORRI
Percentage of the Applicable Percentage secured by (in a contemporaneous
exchange of properties) or reserved to (e.g., in the form of an override or
back-in) Grantor as a result of such transaction, to the extent only the
properties exchanged or the interests reserved are allocable to the Subject
Reserves out of which the subject Term ORRI is carved. Where requested by the
purchaser, farmee, exchange counterparty or transferee, Grantee shall join in
any instrument of sale, farm-out, exchange or transfer.

         (B) In the event any of the Hydrocarbon Interests not constituting
Subject Reserves at the time of sale, farm-out or other transfer (the
"NonSubject Reserves") are sold, farmed out, or otherwise transferred by the
Grantor in a bona fide transaction with a third party that is not an affiliate
of Grantor, the Term ORRI conveyed herein in the NonSubject Reserves so sold,
farmed-out or otherwise transferred shall automatically terminate and merge
into the interest so sold, farmed-out or otherwise transferred upon the final
closing of such transaction; and the Grantee shall not be entitled to any of
the cash proceeds or other consideration received by Grantor on account of the
sale, farm-out or other transfer (either as an Applied Payment or otherwise).
When requested by the purchaser, farmee or transferee on NonSubject Reserves,
Grantee shall join in any instrument of sale, farm-out exchange or transfer.

                                  ARTICLE IV.

          MARKETING OF ORRI HYDROCARBONS AND DISTRIBUTION OF PROCEEDS

         Section 4.1       Nature of Marketing Arrangements. Grantor shall have
                           the obligation to market, or cause to be marketed,
                           the oil, gas and other minerals attributable to the
                           Term ORRI (the "ORRI Hydrocarbons") on behalf of and
                           for the account of Grantee in arm's-length
                           transactions with purchasers who are not Affiliates,
                           on the same terms upon which Grantor markets its own
                           Subject Reserves from the same Subject Well. Grantor
                           shall have no express or implied obligation to sell
                           or otherwise to market or render marketable or more
                           valuable the ORRI Hydrocarbons on any terms better
                           or otherwise different than those pursuant to which
                           it sells, markets or renders marketable or more
                           valuable its own Subject Reserves from the same
                           Subject Well. As to any third parties, all acts of
                           Grantor in marketing the ORRI Hydrocarbons and all
                           sales or other marketing agreements executed by
                           Grantor in accordance herewith shall be binding on
                           Grantee and the Term ORRI; it being understood that
                           the right and obligation to

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                           market the ORRI Hydrocarbons is at all times vested
                           in Grantor, and Grantee does not have any such right
                           or obligation or any possessory interest in all or
                           part of the ORRI Hydrocarbons. Accordingly, it shall
                           not be necessary for Grantee to join in any
                           production sales or marketing agreements or any
                           amendments to existing production sales or marketing
                           agreements.

         Section 4.2       Distribution of Funds. Until notified by Grantee to
                           the contrary, Grantor shall receive all payments for
                           (or on account of) ORRI Hydrocarbons and shall, on
                           or before the last day of each calendar month,
                           distribute any such payments received during the
                           previous calendar month, net only of Permissible
                           Charges, to Agent, on behalf of and for further
                           distribution to Grantee, by wire transfer to such
                           account(s) or location(s) as Agent may direct from
                           time to time in writing. Any monies received by
                           Agent for or on account of ORRI Hydrocarbons shall
                           constitute trust funds in Agent's hands. Except as
                           provided otherwise in the Indenture, Grantee shall
                           have the right at all times, upon thirty (30) days'
                           written notice sent to Grantor, to begin receiving
                           payment for (or on account of) all ORRI Hydrocarbons
                           directly from the purchasers thereof or from any
                           other parties obligated to make payment therefor. In
                           the event Grantee exercises its right to receive
                           payment for (or on account of) ORRI Hydrocarbons
                           directly, Grantor shall immediately cause to be
                           prepared and executed such division orders, transfer
                           orders, or instructions in lieu thereof, as Grantee
                           (or any third party) may reasonably require from
                           time to time to cause payments to be made directly
                           to Grantee; in the event that, for any reason,
                           Grantee cannot (or does not) receive such payments
                           directly, the same shall be collected by Grantor and
                           shall constitute trust funds in Grantor's hands, to
                           be immediately paid over to Grantee by wire transfer
                           to such account or location as Grantee may direct
                           from time to time in writing (or by such other form
                           of transfer reasonably specified by Grantee).

         Section 4.3       Notification to Production Purchasers. ANY AND ALL
                           PURCHASERS OF ORRI HYDROCARBONS SHALL HAVE THE RIGHT
                           TO ASSUME THAT THE TERM ORRI IS STILL VALID AND
                           EXISTING - CALCULATED UTILIZING AN ORRI PERCENTAGE
                           OF 4% UNTIL THE OCCURRENCE OF THE RECALCULATION
                           EVENT - UNTIL SUCH TIME AS SUCH PURCHASER IS
                           NOTIFIED IN WRITING BY GRANTEE TO BEGIN MAKING
                           PAYMENTS ON SOME OTHER BASIS. UNTIL RECEIVING SUCH
                           WRITTEN NOTIFICATION, ALL PURCHASERS OF PRODUCTION
                           SHALL CONTINUE MAKING PAYMENTS IN ACCORDANCE WITH
                           THE FOREGOING ASSUMPTIONS (AND OTHERWISE IN
                           ACCORDANCE WITH

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                           THE DIRECTIVES SET OUT IN SECTION 4.2 HEREOF), AND
                           SHALL NOT PLACE THE FUNDS IN SUSPENSE.

         Section 4.4       Ad Valorem and Severance Taxes. For so long as
                           Grantor is receiving payments and making
                           distributions on account of ORRI Hydrocarbons to
                           Grantee under Section 4.2 hereof, Grantor shall, as
                           an administrative convenience to Grantee and to the
                           extent only that funds are on hand on account of
                           sales of ORRI Hydrocarbons, timely pay and discharge
                           (a) all ad valorem taxes assessed against the
                           Subject Reserves (or the Hydrocarbon Interests
                           directly allocable thereto, including the Term ORRI)
                           or any part thereof, and (b) all production,
                           severance, excise and other taxes assessed against,
                           or measured by, the ORRI Hydrocarbons or the value,
                           or proceeds, of the ORRI Hydrocarbons (the "Taxes").
                           Grantor shall have the right to net Taxes payable
                           under this and any other Term ORRI Conveyance on
                           account of the Term ORRI and the ORRI Hydrocarbons
                           hereunder and thereunder against amounts otherwise
                           distributable on account of the sale of ORRI
                           Hydrocarbons under this and any other Term ORRI
                           Conveyance. Nothing contained in this Section 4.4
                           shall, however, create a personal obligation on the
                           part of Grantor to pay any Taxes on behalf of
                           Grantee to the extent amounts in the possession of
                           Grantee on account of the ORRI Hydrocarbons are
                           insufficient to cover the payment of Taxes on
                           account of the Term ORRI and/or the ORRI
                           Hydrocarbons. Without limitation of the generality
                           of the foregoing, in the event that, in Grantor's
                           reasonable discretion, distributions owing to
                           Grantee will be insufficient to fully cover any ad
                           valorem or similar Taxes allocable to the Term ORRI,
                           Grantee agrees to pay the difference to the
                           appropriate taxing authority within twenty (20) days
                           after receipt from Grantor of the applicable ad
                           valorem or similar Tax assessment. Grantor shall
                           have the right to offset against any distributions
                           owing or to become owing to Grantee the full amount
                           of any Taxes so paid on behalf of Grantee. In the
                           event Grantee elects pursuant to Section 4.2 hereof
                           to receive payment on account of ORRI Hydrocarbons
                           directly, it shall be responsible to pay all Taxes
                           allocable to the Term ORRI directly, for its own
                           account.

                                   ARTICLE V.

                              COVENANTS OF GRANTOR

         Grantor hereby covenants for the benefit of Grantee that Neither
Brigham Parent or Grantor, will allow any Subsidiary to transfer any assets to
Quest Resources LLC or Venture Acquisitions, L.P.

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                                  ARTICLE VI.

                            MISCELLANEOUS PROVISIONS

         Section 6.1       Further Assurances. Grantor agrees to execute and
                           deliver to Grantee, and, to the extent it is
                           reasonably within Grantor's power to do so, to cause
                           any third parties to execute and deliver to Grantee,
                           all such other and additional instruments and to do
                           all such further acts and things as may be necessary
                           or appropriate to more fully vest in and assure to
                           Grantee all of the rights, titles, interests,
                           remedies, powers and privileges herein granted or
                           intended so to be, Grantee agrees to execute and
                           deliver to Grantor all such other and additional
                           instruments and to do all such further acts and
                           things as may be necessary or appropriate to more
                           fully vest in and assure to Grantor all of the
                           rights, titles, interests, remedies, powers and
                           privileges herein granted or intended so to be.

         Section 6.2       Termination. Upon Grantor's request, Grantee will
                           confirm to Grantor in writing reasonably acceptable
                           to Grantor the occurrence of the Recalculation Event
                           and the termination of the Term ORRI (as to all or
                           any one or more Hydrocarbon Interests).

         Section 6.3       No Waiver. The failure of any party to insist upon
                           strict performance of a covenant hereunder or of any
                           obligation hereunder, irrespective of the length of
                           time for which such failure continues, shall not be
                           a waiver of such party's right to demand strict
                           compliance in the future. No consent or waiver,
                           express or implied, to or of any breach or default
                           in the performance of any obligation hereunder shall
                           constitute a consent or waiver to or of any other
                           breach or default in the performance of the same or
                           any other obligation hereunder.

         Section 6.4       Applicable Law. THIS CONVEYANCE AND THE RIGHTS AND
                           OBLIGATIONS OF THE PARTIES HEREUNDER SHALL, WITHOUT
                           REGARD TO PRINCIPLES OF CONFLICTS OF LAWS, BE
                           GOVERNED BY AND INTERPRETED, CONSTRUED AND ENFORCED
                           IN ACCORDANCE WITH THE LAWS OF THE STATE OF TEXAS,
                           EXCEPT TO THE EXTENT THE LAWS OF ANY STATE IN WHICH
                           ANY PART OF THE HYDROCARBON INTERESTS IS LOCATED
                           NECESSARILY, GOVERNS WITH RESPECT TO SUCH PART OF
                           THE HYDROCARBON INTERESTS LOCATED IN SUCH STATE.

                                     -11-

<PAGE>   12

         Section 6.5       Severability. Every provision in this Conveyance is
                           intended to be severable. If any term or provision
                           hereof is determined to be invalid, illegal or
                           unenforceable for any reason whatsoever, such
                           invalidity, illegality or unenforceability shall not
                           affect the validity, legality and enforceability of
                           the remainder of this Conveyance.

         Section 6.6       No Personal Liability of Grantee/Indemnities.
                           NOTWITHSTANDING ANYTHING TO THE CONTRARY CONTAINED
                           IN THIS CONVEYANCE, GRANTEE SHALL NEVER PERSONALLY
                           BE RESPONSIBLE FOR PAYMENT OF ANY PART OF THE COSTS,
                           EXPENSES OR LIABILITIES INCURRED IN CONNECTION WITH
                           THE EXPLORING, DEVELOPING, OPERATING, OWNING AND/OR
                           MAINTAINING OF THE HYDROCARBON INTERESTS. GRANTOR
                           AGREES TO INDEMNIFY AND HOLD GRANTEE AND ITS
                           DIRECTORS, OFFICERS, EMPLOYEES, REPRESENTATIVES,
                           AGENTS AND AFFILIATES, HAS FROM AND AGAINST ALL SUCH
                           COSTS, EXPENSES AND LIABILITIES (WITH SUCH INDEMNITY
                           TO ALSO COVER ALL COSTS AND EXPENSES OF GRANTEE,
                           INCLUDING REASONABLE LEGAL FEES AND EXPENSES, WHICH
                           ARE INCURRED INCIDENT TO THE MATTERS INDEMNIFIED
                           AGAINST); PROVIDED, HOWEVER; THAT PERMISSIBLE
                           CHARGES SHALL, TO THE EXTENT THE SAME RELATE TO
                           PERIODS AFTER THE EFFECTIVE DATE, NEVERTHELESS BE
                           CHARGEABLE AGAINST THE TERM ORRI OR OTHERWISE
                           PAYABLE AND/OR BORNE DIRECTLY BY GRANTEE UNDER
                           CIRCUMSTANCES DESCRIBED HEREIN, AS AND TO THE EXTENT
                           HEREIN PERMITTED. EXPRESSLY WITHOUT LIMITATION OF
                           THE GENERALITY OF THE FOREGOING, GRANTOR HEREBY
                           AGREES TO INDEMNIFY AND HOLD GRANTEE AND ITS
                           DIRECTORS, OFFICERS, EMPLOYEES, REPRESENTATIVES,
                           AGENTS AND AFFILIATES, HARMLESS FROM AND AGAINST ALL
                           CLAIMS, DEMANDS, LIABILITIES, LOSSES, DAMAGES, COSTS
                           AND EXPENSES (A) ARISING FROM ANY FAILURE OF GRANTOR
                           TO PERFORM OR COMPLY WITH ANY OF THE PROVISIONS OF
                           THIS CONVEYANCE, (B) ARISING OUT OF THE CONDITION OF
                           THE HYDROCARBON INTERESTS (WHETHER KNOWN OR UNKNOWN,
                           LATENT OR PATENT) ON, BEFORE OR AFTER THE DATE
                           HEREOF, AND/OR (C) ARISING UNDER ANY ENVIRONMENTAL
                           LAWS, WHETHER NOW IN EXISTENCE OR HEREINAFTER

                                     -12-

<PAGE>   13

                           ENACTED. THE INDEMNITIES AND HOLD HARMLESS
                           PROVISIONS CONTAINED IN THIS SECTION 7.6 SHALL APPLY
                           WHETHER OR NOT THE INDEMNIFIED PARTY WAS WHOLLY OR
                           PARTIALLY NEGLIGENT (OTHER THAN GROSSLY NEGLIGENT).

         Section 6.7       Counterparts. This Conveyance is being executed in
                           several counterparts, all of which are identical,
                           except that, to facilitate recordation, in certain
                           counterparts hereof only that portion of Exhibit A
                           which contains specific descriptions of the
                           Hydrocarbon Interests located in the recording
                           jurisdiction in which the counterpart is to be
                           recorded shall be included, and all other portions
                           of Exhibit A shall be included by reference only.
                           All of such counterparts together shall constitute
                           one and the same instrument. Complete copies of this
                           Conveyance, containing the entire Exhibit A, have
                           been retained by Grantor and Grantee.

                                     -13-

<PAGE>   14

         THIS CONVEYANCE IS EXECUTED this 17th day of February, 2000, and made
effective as of the Effective Date.

                                       BRIGHAM OIL & GAS, L.P.

                                       By:  Brigham, Inc. a Nevada corporation,
                                            its General Partner

                                            By:
                                                -------------------------------
                                                Karen E. Lynch
                                                Vice President

ADDRESS OF GRANTOR:                         ADDRESS OF GRANTEE:
6300 Bridge Point Parkway                   1400 Smith Street
Austin, Texas 78730                         Houston, Texas 77002

This instrument prepared by:
Gray H. Muzzy
Bracewell & Patterson, L.L.P.
711 Louisiana
Houston, Texas 77002
(713) 221-1406

<PAGE>   15

                                 ACKNOWLEDGMENT

STATE OF TEXAS                    )
                                  )
COUNTY OF HARRIS                  )

                                     KANSAS

         The foregoing instrument was acknowledged before me on February 17,
2000, by Karen Lynch, Vice President of Brigham, Inc., a Nevada corporation, as
general partner of Brigham Oil & Gas, L.P., a Delaware limited partnership, on
behalf of the corporation, as general partner of the limited partnership.

                                   LOUISIANA

         This instrument was acknowledged by me on February 17, 2000 by Karen
Lynch, Vice President of Brigham, Inc., a Nevada corporation, as general
partner of Brigham Oil & Gas, L.P., a Delaware limited partnership, on behalf
of the corporation, as general partner of the limited partnership.

                                    MONTANA

         The foregoing instrument was acknowledged before me on February 17,
2000, by Karen Lynch, Vice President of Brigham, Inc., a Nevada corporation, as
general partner of Brigham Oil & Gas, L.P., a Delaware limited partnership, on
behalf of the corporation, as general partner of the limited partnership.

                                   NEW MEXICO

         The foregoing instrument was acknowledged before me on February 17,
2000, by Karen Lynch, Vice President of Brigham, Inc., a Nevada corporation, as
general partner of Brigham Oil & Gas, L.P., a Delaware limited partnership, on
behalf of the corporation, as general partner of the limited partnership.

                                    OKLAHOMA

         Before me, a Notary Public in and for said county and state, on this
17th day of February, 2000, personally appeared Karen Lynch, to me know to be
the identical person who subscribed the name of the maker thereof to the
foregoing instrument as Vice President of BRIGHAM, INC., as general partner of
BRIGHAM OIL & GAS, L.P., a Delaware limited partnership, and acknowledged to me
that she executed the same as her free and voluntary act and deed, and as the
free and voluntary act and deed of such corporation for the uses and purposes
therein set forth.

<PAGE>   16

                                     TEXAS

         This instrument was acknowledged before me on February 17, 2000 by
Karen E. Lynch, Vice President of BRIGHAM, INC., a Nevada corporation, as
general partner of BRIGHAM OIL & GAS, L.P., a Delaware limited partnership, on
behalf of such corporation, as general partner of the limited partnership.

                                       ----------------------------------------
                                       NOTARY PUBLIC IN AND FOR
                                       THE STATE OF TEXAS

                                       Notarial Seal:

<PAGE>   17

                                   EXHIBIT A

            [Describe leases and wells - to be provided by Brigham]

<PAGE>   18

                                   EXHIBIT B

                                  TITLES, ETC.

 (a) The rights and interests afforded Gasco Limited Partnership ("Gasco") and
its successors and assigns under the terms of (i) that certain Expense
Allocation and Participation Agreement dated as of April 1, 1996, by and
between Gasco and Grantor, as heretofore and/or hereafter amended, expanded,
supplemented, renewed and/or extended, and (ii) that certain Expense Allocation
and Participation Agreement II dated as of April 1, 1997, by and between Gasco
and Grantor, as heretofore and/or hereafter amended, expanded, supplemented,
renewed and/or extended, and (iii) that certain Expense Allocation and
Participation Agreement III dated as of March 1, 1998, by and between Gasco and
Grantor, as heretofore amended, expanded, supplemented, renewed and/or
extended.

(b) The rights and interests afforded Middle Bay Oil Company ("Middle Bay") and
its successors and assigns under the terms of that certain Expense Allocation
and Participation Agreement dated as of April 1, 1996, by and between Middle
Bay and Grantor, as heretofore amended, modified, expanded, supplemented,
renewed and/or extended.

(c) The rights and interests afforded Stephens Production Company ("Stephens")
and its successors and assigns under the terms of that certain Anadarko Basin
Joint Participation Agreement dated as of May 1, 1996, by and between Grantor
and Stephens, as heretofore amended, modified, expanded, supplemented, renewed
and/or extended.

(d) The rights and interests afforded Vintage Petroleum, Inc. ("Vintage") and
its successors and assigns under the terms of that certain Anadarko Basin Joint
Participation Agreement dated as of May 1, 1996, by and between Grantor and
Vintage, as heretofore amended, modified, expanded, supplemented, renewed
and/or extended.

(e) The rights and interests afforded Brigham-Duke (Delaware) LLC ("Duke") and
its successors and assigns under the terms of that certain Joint Development
Agreement dated as of February 17, 1999, by and between Duke and Grantor, as
heretofore amended.

(f) The rights and interests, whether real or equitable, vested or contingent,
afforded third parties under the terms of various farmout, exploration, joint
participation and/or operating agreements, in existence as of the date hereof,
under or in relation to which Grantor, Brigham Parent or any Subsidiary agrees
to farm-out to one or more third party(ies) one or more Oil and Gas Property
and/or undertakes with one or more third party(ies) the joint exploration
and/or development of one or more Oil and Gas Properties (which rights and
interests may include, without limitation, rights under Area of Mutual
Interests agreements and rights in respect of forfeiture of all or a portion of
Brigham Parent's interests in an Oil and Gas Property, or part thereof or
interest therein, that is triggered by an election not to participate in a
proposed operation; and

<PAGE>   19

(g) consulting agreements in existence as of the date hereof, as same may have
heretofore been amended, expanded, supplemented, renewed and/or extended, with
third party geologists, landmen or other oil and gas industry participants who
agree to perform services in return, in whole or in part, for an overriding
royalty interest or other interest in or relating to any Oil and Gas
Properties.<PAGE>   1
                                                                  EXHIBIT 10.13

                               WARRANT CERTIFICATE

Number of Warrants: 750,000                                 Warrant No.  A-6

         This warrant certificate ("Warrant Certificate") certifies that, for
value received,

         JOINT ENERGY DEVELOPMENT INVESTMENTS II LIMITED PARTNERSHIP

is the registered holder of the number of warrants (the "Warrants") set forth
above. Each Warrant entitles the holder thereof, at any time or from time to
time during the Exercise Period, to purchase from the Company one fully paid and
nonassessable share of Common Stock at the Exercise Price, subject to adjustment
as provided herein. The Warrants constitute, as of February 17, 2000, 3.1015% of
the outstanding Common Stock on a fully diluted basis including, for purposes of
such calculation, the Acquired Shares and the Warrant Shares. Initially
capitalized terms used but not defined herein shall have the meanings ascribed
to them in the Securities Purchase Agreement. This Warrant Certificate amends
and restates Warrant No. A-4 issued to Joint Energy Development Investments II
Limited Partnership as of March 26, 1999, which in turn had amended and restated
Warrant No. A-2 issued to Joint Energy Development Investments II Limited
Partnership on August 20, 1998.

         "Acquired Shares" means the shares of Common Stock acquired by Enron
North America Corp. and Joint Energy Development Investments II Limited
Partnership pursuant to Section 2.01 of the Securities Purchase Agreement.

         "Appraiser" has the meaning set forth in Section 13.

         "Appreciation Amount" has the meaning set forth in Section 1(b).

         "Common Stock" means the common stock, $.01 par value per share, of the
Company and such other class of securities as shall then represent the common
equity of the Company.

         "Company" means Brigham Exploration Company, a Delaware corporation.

         "Exercise Period" means the period of time between the Funding Date, as
defined in the Securities Purchase Agreement and 5:00 p.m. (New York City time)
on the Expiration Date.

         "Exercise Price" subject in all circumstances to adjustment in
accordance with Section 2, means $______, which represents the average volume
weighted Price during the twenty (20) calendar day period that begins February
22, 2000 and ends March 12, 2000.

         "Expiration Date" means August 22, 2008.

<PAGE>   2

         "Dissenting Notice" has the meaning set forth in Section 13.

         "Fair Market Value" has the meaning set forth in Section 13.

         "Funding Date" is defined in the Securities Purchase Agreement.

         "IPO" shall mean the initial public offering of securities of the
Company consummated on May 24, 1997, pursuant to a registration statement filed
under the Securities Act of 1933, as amended, and the rules and regulations
promulgated thereunder.

         "Issuance Date" means August 20, 1998.

         "Notes" means the Senior Subordinated Secured Notes issued pursuant to
the Securities Purchase Agreement.

         "Person" means any individual, corporation, company, partnership, joint
venture, trust, limited liability company, unincorporated organization or
government or any agency, instrumentality or political subdivision thereof, or
any other form of entity.

         "Price" means the average of the "high" and "low" prices as reported in
The Wall Street Journal's listing for such day (corrected for obvious
typographical errors) or if such shares are not reported in such listing, the
average of the reported "high" and "low" sales prices on the largest national
securities exchange (based on the aggregate dollar value of securities listed)
on which such shares are listed or traded, or if such shares are not listed or
traded on any national securities exchange, then the average of the reported
"high" and "low" sales prices for such shares in the over- the-counter market,
as reported on the National Association of Securities Dealers Automated
Quotations System, or, if such prices shall not be reported thereon, the average
of the closing bid and asked prices so reported, or, if such prices shall not be
reported, then the average of the closing bid and asked prices reported by the
National Quotations Bureau Incorporated, or, in all other cases, the Fair Market
Value as determined in accordance with Section 13 below. The "average" Price per
share for any period shall be determined by dividing the sum of the Prices
determined for the individual trading days in such period by the number of
trading days in such period.

         "Publicly Traded" means, with respect to the Common Stock, that such
securities are listed for trading on the New York Stock Exchange, Inc., the
American Stock Exchange, Inc. or the NASDAQ National Market. The Common Stock
shall also be deemed to be Publicly Traded if the Common Stock is included in
the NASDAQ SmallCap Market.

         "Securities Purchase Agreement" means the Securities Purchase
Agreement, dated as of August 20, 1998, among the Company, Enron North America
Corp. (formerly named Enron Capital

                                      -2-
<PAGE>   3

& Trade Resources Corp.), and Joint Energy Development Investments II Limited
Partnership, individually and as agent, as amended from time to time.

         "Warrant Shares" means the shares of Common Stock and other securities
receivable upon exercise of the Warrants.

         "Warrants" means the Warrants issued by the Company to Enron North
America Corp. and Joint Energy Development Investments II Limited Partnership
pursuant to the Securities Purchase Agreement and any Warrants issued upon the
transfer thereof or in substitution therefor.

         1. EXERCISE OF WARRANTS. (a) The Warrants may be exercised in whole or
in part, at any time or from time to time, during the Exercise Period, by
presentation and surrender to the Company at its address set forth in Section 9
of (i) this Warrant Certificate with the Election To Exercise, attached hereto
as Exhibit A-1, duly completed and executed, and (ii) payment of the Exercise
Price for the number of Warrants being exercised. At the option of the holder
hereof, payment of the Exercise Price shall be made (w) by wire transfer of
funds, (x) by check payable to the order of the Company, (y) by application of
any Warrant Shares or credit against any Notes, as provided below, or (z) by any
combination of such methods.

If the holder of this Warrant Certificate at any time exercises less than all
the Warrants, the Company shall issue to such holder a warrant certificate
identical in form to this Warrant Certificate, but evidencing a number of
Warrants equal to the number of Warrants originally represented by this Warrant
Certificate less the number of Warrants previously exercised. Likewise, upon the
presentation and surrender of this Warrant Certificate to the Company at its
address set forth in Section 9 and at the request of the holder, the Company
will, without expense, at the option of the holder, issue to the holder in
substitution for this Warrant Certificate one or more warrant certificates in
identical form and for an aggregate number of Warrants equal to the number of
Warrants evidenced by this Warrant Certificate.

              (b) In lieu of exercise in accordance with paragraph (a) above,
the Warrants represented hereby may be exercised, in whole or in part, by
presentation and surrender at the office of the Company specified herein of this
Warrant Certificate with the "Alternative Election to Exercise", attached hereto
as Exhibit A-2, duly completed and executed. Upon such exercise, the holder
shall be entitled to receive from the Company, for each share of Common Stock
issuable upon exercise of each Warrant being exercised, shares of Common Stock
with a value equal to the amount by which either the Price or, in the event that
the Common Stock is not Publicly Traded, the Fair Market Value of one share of
Common Stock on the date of exercise exceeds the Exercise Price (the
"Appreciation Amount"). If the Common Stock is Publicly Traded on the date

                                      -3-
<PAGE>   4

of exercise of Warrants pursuant to this Section 1(b), the Company shall deliver
(or cause its transfer agent to deliver) certificates representing shares of
Common Stock issuable upon such exercise within five business days following the
date of exercise. If the Common Stock is not Publicly Traded on the date of
exercise of Warrants pursuant to this Section 1(b), such election may be
withdrawn by the holder at any time prior to the third business day following
the date on which the holder receives notice of the final determination of Fair
Market Value pursuant to Section 15. Unless the holder timely withdraws its
exercise, the Company shall deliver (or cause its transfer agent to deliver)
certificates representing Common Stock so issuable within 10 business days
following the date of determination of the Fair Market Value. In lieu of any
fractional shares otherwise issuable upon exercise pursuant hereto, the Company
shall pay to the holder cash in an amount equal to the Price, or in the event
the Company is not Publicly Traded, the Fair Market Value of such fractional
share.

              (c) Upon the exercise of this Warrant in whole or in part by the
holder of any Notes, such holder may, at its option, deliver written notice to
the Company that the holder has elected to apply some or all of the Exercise
Price required upon such exercise against all or any specified principal amount
of and/or accrued but unpaid interest on, such Notes, in which case the Company
will accept the application of the Exercise Price against such specified
principal and/or interest amount in lieu of a like amount of cash payment.

              (d) To the extent that the Warrants have not been exercised at or
prior to the Expiration Date, such Warrants shall expire and the rights of the
holder shall become void and of no effect.

         2. ANTIDILUTION ADJUSTMENTS. The shares of Common Stock purchasable on
exercise of the Warrants are shares of Common Stock as constituted as of the
Issuance Date. The number and kind of securities purchasable upon the exercise
of the Warrants, and the Exercise Price, shall be subject to adjustment from
time to time upon the happening of certain events, as follows:

              (a) Mergers, Consolidations and Reclassifications. In case of any
reclassification or change of outstanding securities issuable upon exercise of
the Warrants at any time after the Issuance Date (other than a change in par
value, or from par value to no par value, or from no par value to par value or
as a result of a subdivision or combination to which subsection 2(b) applies),
or in case of any consolidation or merger of the Company with or into another
entity or other person (other than a merger with another entity or other person
in which the Company is the surviving corporation and which does not result in
any reclassification or change in the securities issuable upon exercise of this
Warrant Certificate), the holder of the Warrants shall have, and the Company, or
such successor corporation or other entity, shall covenant in the constituent
documents effecting any of the foregoing transactions that such holder does
have, the right to obtain upon the exercise of the Warrants, in lieu of each
share of Common Stock, other securities, money or other property theretofore
issuable upon exercise of a Warrant, the kind and amount of shares of stock,
other securities, money or other property receivable upon such

                                      -4-
<PAGE>   5

reclassification, change, consolidation or merger by a holder of the shares of
Common Stock, other securities, money or other property issuable upon exercise
of a Warrant if the Warrants had been exercised immediately prior to such
reclassification, change, consolidation or merger. The constituent documents
effecting any such reclassification, change, consolidation or merger shall
provide for adjustments which shall be as nearly equivalent as may be
practicable to the adjustments provided in this subsection 2(a). The provisions
of this subsection 2(a) shall similarly apply to successive reclassifications,
changes, consolidations or mergers.

         (b) Subdivisions and Combinations. If the Company, at any time after
the Issuance Date, shall subdivide its shares of Common Stock into a greater
number of shares, the Exercise Price in effect immediately prior to such
subdivision shall be proportionately reduced, and the number of shares of Common
Stock purchasable upon exercise of the Warrants shall be proportionately
increased, as at the effective date of such subdivision, or if the Company shall
take a record of holders of its Common Stock for such purpose, as at such record
date, whichever is earlier. If the Company, at any time after the Issuance Date,
shall combine its shares of Common Stock into a smaller number of shares, the
Exercise Price in effect immediately prior to such combination shall be
proportionately increased, and the number of shares of Common Stock purchasable
upon exercise of the Warrants shall be proportionately reduced, as at the
effective date of such combination, or if the Company shall take a record of
holders of its Common Stock for purposes of such combination, as at such record
date, whichever is earlier.

         (c) Dividends and Distributions. If the Company at any time after the
Issuance Date shall declare a dividend on its Common Stock payable in stock or
other securities of the Company to the holders of its Common Stock, the holder
of this Warrant Certificate shall, without additional cost, be entitled to
receive upon any exercise of a Warrant, in addition to the Common Stock to which
such holder would otherwise be entitled upon such exercise, the number of shares
of stock or other securities which such holder would have been entitled to
receive if he had been a holder immediately prior to the record date for such
dividend (or, if no record date shall have been established, the payment date
for such dividend) of the number of shares of Common Stock purchasable on
exercise of such Warrant immediately prior to such record date or payment date,
as the case may be.

         (d) Certain Issuances of Securities. If the Company at any time after
the Issuance Date shall issue any additional shares of Common Stock (otherwise
than as provided in paragraphs (a) through (c) of this Section 2) at a price per
share less than the average Price per share of Common Stock for the 20 trading
days immediately preceding the date of the authorization of such issuance (the
"Market Price") by the Board of Directors, then the Exercise Price upon each
such issuance shall be adjusted to that price determined by multiplying the
Exercise Price by a fraction:

              i. the numerator of which shall be the sum of (1) the number of
     shares of Common Stock outstanding immediately prior to the issuance of
     such additional shares of Common Stock multiplied by the Market Price, and
     (2) the consideration, if any, received by the Company upon the issuance of
     such additional shares of Common Stock, and

                                      -5-
<PAGE>   6

              ii. the denominator of which shall be the Market Price multiplied
     by the total number of shares of Common Stock outstanding immediately after
     the issuance of such additional shares of Common Stock.

     No adjustments of the Exercise Price shall be made under this paragraph (d)
upon the issuance of any additional shares of Common Stock that (y) are issued
pursuant to thrift plans, stock purchase plans, stock bonus plans, stock option
plans, employee stock ownership plans and other incentive or profit sharing
arrangements for the benefit of employees ("Employee Benefit Plans") that
otherwise would cause an adjustment under this paragraph (d); provided that the
aggregate number of shares of Common Stock so issued (including the shares
issued pursuant to any options, rights or warrants or convertible or
exchangeable securities issued under such Employee Benefit Plans containing the
right to purchase shares of Common Stock) pursuant to Employee Benefit Plans
after the closing date of the IPO, as adjusted for any stock splits, stock
dividends or subdivisions or combinations of Common Stock prior to the
Expiration Date, shall not in the aggregate exceed 5% of the Company's
outstanding Common Stock at the time of such issuance; or (z) are issued
pursuant to any Common Stock Equivalent (as hereinafter defined) (i) if upon the
issuance of any such Common Stock Equivalent, any such adjustments shall
previously have been made pursuant to paragraph (e) of this Section 2 or (ii) if
no adjustment was required pursuant to paragraph (e) of this Section 2.

         (e) Common Stock Equivalents. If the Company shall, after the Issuance
Date, issue any security or evidence of indebtedness which is convertible into
or exchangeable for Common Stock ("Convertible Security"), or any warrant,
option or other right to subscribe for or purchase Common Stock or any
Convertible Security, other than pursuant to Employee Benefit Plans (together
with Convertible Securities, "Common Stock Equivalent"), or if, after any such
issuance, the price per share for which additional shares of Common Stock may be
issuable thereunder is amended, then the Exercise Price upon each such issuance
or amendment shall be adjusted as provided in subsection (d) on the basis that
(i) the maximum number of additional shares of Common Stock issuable pursuant to
all such Common Stock Equivalents shall be deemed to have been issued as of the
earlier of (a) the date on which the Company shall enter into a firm contract
for the issuance of such Common Stock Equivalent, or (b) the date of actual
issuance of such Common Stock Equivalent; and (ii) the aggregate consideration
for such maximum number of additional shares of Common Stock shall be deemed to
be the minimum consideration received and receivable by the Company for the
issuance of such additional shares of Common Stock pursuant to such Common Stock
Equivalent; provided, however, that no adjustment shall be made pursuant to this
subsection (e) unless the consideration received and receivable by the Company
per share of Common Stock for the issuance of such additional shares of Common
Stock pursuant to such Common Stock Equivalent is less than the Market Price. No
adjustment of the Exercise Price shall be made under this subsection (e) upon
the issuance of any Convertible Security which is issued pursuant to the
exercise of any warrants or other subscription or purchase rights therefor, if
any adjustment shall previously have been made in the Exercise Price then in
effect upon the issuance of such warrants or other rights pursuant to this
subsection (e).

                                      -6-
<PAGE>   7

         (f) Miscellaneous. The following provisions shall be applicable to the
making of adjustments in the Exercise Price hereinbefore provided in this
Section 2:

              i. The consideration received by the Company shall be deemed to be
     the following: (I) to the extent that any additional shares of Common Stock
     or any Common Stock Equivalent shall be issued for cash consideration, the
     consideration received by the Company therefor, or, if such additional
     shares of Common Stock or Common Stock Equivalent are offered by the
     Company for subscription, the subscription price, or, if such additional
     shares of Common Stock or Common Stock Equivalent are sold to underwriters
     or dealers for public offering without a subscription offering, the initial
     public offering price, in any such case excluding any amounts paid or
     receivable for accrued interest or accrued dividends and without deduction
     of any compensation, discounts, commissions or expenses paid or incurred by
     the Company for and in the underwriting of, or otherwise in connection
     with, the issue thereof; (II) to the extent that such issuance shall be for
     a consideration other than cash, then, except as herein otherwise expressly
     provided, the fair value of such consideration at the time of such issuance
     as determined in good faith by the Board of Directors, as evidenced by a
     certified resolution of the Board of Directors delivered to the holder of
     this Warrant Certificate setting forth such determination. The
     consideration for any additional shares of Common Stock issuable pursuant
     to any Common Stock Equivalent shall be the consideration received by the
     Company for issuing such Common Stock Equivalent, plus the additional
     consideration payable to the Company upon the exercise, conversion or
     exchange of such Common Stock Equivalent. In case of the issuance at any
     time of any additional shares of Common Stock or Common Stock Equivalent in
     payment or satisfaction of any dividend upon any class of stock other than
     Common Stock, the Company shall be deemed to have received for such
     additional shares of Common Stock or Common Stock Equivalent (which shall
     not be deemed to be a dividend payable in, or other distribution of, Common
     Stock under subsection (c) above) consideration equal to the amount of such
     dividend so paid or satisfied.

              ii. Upon the expiration of the right to convert, exchange or
     exercise any Common Stock Equivalent the issuance of which effected an
     adjustment in the Exercise Price, if any such Common Stock Equivalent shall
     not have been converted, exercised or exchanged, the number of shares of
     Common Stock deemed to be issued and outstanding because they were issuable
     upon conversion, exchange or exercise of any such Common Stock Equivalent
     shall no longer be computed as set forth above, and the Exercise Price
     shall forthwith be readjusted and thereafter be the price which it would
     have been (but reflecting any other adjustments in the Exercise Price made
     pursuant to the provisions of subsection (d) after the issuance of such
     Common Stock Equivalent) had the adjustment of the Exercise Price made upon
     the issuance or sale of such Common Stock Equivalent been made on the basis
     of the issuance only of the

                                      -7-
<PAGE>   8

     number of additional shares of Common Stock actually issued upon exercise,
     conversion or exchange of such Common Stock Equivalent and thereupon only
     the number of additional shares of Common Stock actually so issued shall be
     deemed to have been issued and only the consideration actually received by
     the Company (computed as in subparagraph (i) of this paragraph (f)) shall
     be deemed to have been received by the Company.

              iii. The number of shares of Common Stock at any time outstanding
     shall not include any shares thereof then directly or indirectly owned or
     held by or for the account of the Company or its wholly owned subsidiaries.

              iv. For the purposes of this Section 2, the term "shares of Common
     Stock" shall mean shares of (i) the class of stock designated as the Common
     Stock at the date hereof or (ii) any other class of stock resulting from
     successive changes or reclassifications of such shares consisting solely of
     changes in par value, or from par value to no par value, or from no par
     value to par value. If at any time, because of an adjustment pursuant to
     subsection (a), the Warrants shall entitle the holders to purchase any
     securities other than shares of Common Stock, thereafter the number of such
     other securities so purchasable upon exercise of each Warrant and the
     Exercise Price of such securities shall be subject to adjustment from time
     to time in a manner and on terms as nearly equivalent as practicable to the
     provisions with respect to the Common Stock contained in this Section 2.

         (g) Calculation of Exercise Price. The Exercise Price in effect from
time to time shall be calculated to four decimal places and rounded to the
nearest thousandth.

     3. NOTICE OF ADJUSTMENTS. Whenever the Exercise Price or the number of
shares of Common Stock is required to be adjusted as provided in Section 2, the
Company shall forthwith compute the adjusted Exercise Price or the number of
shares of Common Stock issuable and shall prepare and mail to the holder hereof
a certificate setting forth such adjusted Exercise Price or such number of
shares of Common Stock, showing in reasonable detail the facts upon which the
adjustment is based.

     4. VOLUNTARY REDUCTION. (a) The Company may at its option, but shall not be
obligated to, at any time during the term of the Warrants, reduce the then
current Exercise Price by any amount selected by the Board of Directors;
provided that if the Company elects so to reduce the then current Exercise
Price, such reduction shall be irrevocable during its effective period and
remain in effect for a minimum of 30 days following the date of such election,
after which time the Company may, at its option, reinstate the Exercise Price in
effect prior to such reduction. Whenever the Exercise Price is reduced, the
Company shall mail to the holder a notice of the reduction at least 30 days
before the date the reduced Exercise Price takes effect, stating the reduced
Exercise Price and the period for which such reduced Exercise Price will be in
effect.

                                      -8-
<PAGE>   9

         (b) The Company may make such decreases in the Exercise Price, in
addition to those required or allowed by this Section 4, as shall be determined
by it, as evidenced by a certified resolution of the Board of Directors
delivered to the holders, to be advisable to avoid or diminish any income tax to
the holder resulting from any dividend or distribution of stock or issuance of
rights or warrants to purchase or subscribe for stock or from any event treated
as such for income tax purposes.

     5. NOTICES TO WARRANT HOLDERS. In the event:

         (a) the Company shall authorize any consolidation or merger to which
the Company is a party and for which approval of any stockholders of the Company
is required, or of the conveyance or sale of all or substantially all of the
assets of the Company, or of any reclassification or change of the Common Stock
or other securities issuable upon exercise of the Warrants (other than a change
in par value, or from par value to no par value, or from no par value to par
value or as a result of a subdivision or combination), or a tender offer or
exchange offer for shares of Common Stock (or other securities issuable upon the
exercise of the Warrants); or

         (b) the Company shall declare any dividend (or any other distribution)
on the Common Stock or any other class of its capital stock; or

         (c) the Company shall authorize the granting to the holders of Common
Stock or any other class of its capital stock of rights or warrants to subscribe
for or purchase any shares of any class or series of capital stock or any other
securities convertible into or exchangeable for shares of stock; or

         (d) of the voluntary or involuntary dissolution, liquidation or winding
up of the Company;

     then the Company shall cause to be sent to the holder hereof, at least 30
days prior to the applicable record date hereinafter specified, or promptly in
the case of events for which there is no record date, a written notice stating
(x) the date for the determination of the holders of record of shares of Common
Stock (or other securities issuable upon the exercise of the Warrants) entitled
to receive any such dividends or other distribution, (y) the initial expiration
date set forth in any tender offer or exchange offer for shares of Common Stock
(or other securities issuable upon the exercise of the Warrants), or (z) the
date on which any of the events specified in subsections (a)-(d) is expected to
become effective or consummated, and the date as of which it is expected that
holders of record of shares of Common Stock (or other securities issuable upon
the exercise of the Warrants) shall be entitled to exchange such shares for
securities or other property, if any, deliverable upon any such event. Failure
to give such notice or any defect therein shall not affect the legality or
validity of any such event, or the vote upon any such action.

                                      -9-
<PAGE>   10

     6. REPORTS TO WARRANT HOLDERS. The Company will cause to be delivered, by
first-class mail, postage prepaid, to the holder at such holder's address
appearing hereon, or such other address as the holder shall specify, a copy of
any reports delivered by the Company to the holders of Common Stock.

     7. COVENANTS OF THE COMPANY. The Company covenants and agrees that:

              (a) Until the Expiration Date, the Company shall at all times
reserve and keep available, free from preemptive rights, out of the aggregate of
its authorized but unissued Common Stock (and other securities), for the purpose
of enabling it to satisfy any obligation to issue shares of Common Stock (and
other securities) upon the exercise of the Warrants, the number of shares of
Common Stock (and other securities) issuable upon the exercise of such Warrants.

              (b) The Company shall pay all expenses, taxes and other charges
payable in connection with the preparation, issuance and delivery of new warrant
certificates on transfer of the Warrants.

              (c) All Common Stock (and other securities) which may be issued
upon exercise of the Warrants shall upon issuance be validly issued, fully paid,
non-assessable and free from all preemptive rights and all taxes, liens and
charges with respect to the issuance thereof, and will not be subject to any
restrictions on voting or transfer thereof except as set forth in any
stockholders agreement.

              (d) All original issue taxes payable in respect of the issuance of
shares of Common Stock to the registered holder hereof upon the exercise of the
Warrants shall be borne by the Company; provided, that the Company shall not be
required to pay any tax or charge imposed in connection with any transfer
involved in the issuance of any certificate representing shares of Common Stock
(and other securities) in any name other than that of the registered holder
hereof, and in such case the Company shall not be required to issue or deliver
any certificate representing shares of Common Stock (and other securities) until
such tax or other charge has been paid or it has been established to the
Company's satisfaction that no such tax or charge is due.

              (e) As soon as practicable after the receipt from the holder of
this Warrant Certificate of notice of the exercise of a number of warrants
sufficient to require a filing under the Hart-Scott-Rodino Antitrust
Improvements Act of 1976 and the rules, regulations and formal interpretations
thereunder, as amended from time to time (the "HSR Act"), but in any event no
later than the 10th business day after receipt of such notice, the Company will
(i) prepare and file with the Antitrust Division of the Department of Justice
(the "DOJ") and the Federal Trade Commission (the "FTC") the Notification and
Report Form (accompanied by all documentary attachments contemplated thereby)
required by the HSR Act, (ii) upon the request of the holder, request early
termination of the waiting period imposed by the HSR Act, (iii) coordinate and
cooperate with the

                                      -10-
<PAGE>   11

holder in responding to formal and informal requests for additional information
and documentary material from the DOJ and the FTC in connection with such
filing, (iv) use its best efforts to take, or cause to be taken, all reasonable
action and to do, or cause to be done, all things necessary and appropriate to
permit the issuance to the holder of the shares of Common Stock issuable upon
the exercise of the warrants with respect to which any filing is required under
the HSR Act, and (v) reimburse the holder for the entire amount of any filing
fee or any other costs and expenses incurred by the holder in connection
therewith (including legal fees), or as required to be paid under the HSR Act.

              (f) QUOTATION ON NASDAQ. The Company shall maintain the
designation and quotations, or listing, of its Common Stock on the NASDAQ
national market (or on the New York Stock Exchange or the American Stock
Exchange) until the date on which none of the Warrants or Warrant Shares remain
outstanding.

     8. NO RIGHTS AS STOCKHOLDER. The holder of the Warrants shall not, by
virtue of holding such Warrants, be entitled to any rights of a stockholder of
the Company either at law or in equity, and the rights of the holder of the
Warrants are limited to those expressed herein.

     9. NOTICES. All notices provided for hereunder shall be in writing and may
be given by registered or certified mail, return receipt requested, telex,
telegram, telecopier, air courier guaranteeing overnight delivery of personal
delivery, if to the holder at the following address:

        Joint Energy Development Investments II Limited Partnership
        1400 Smith Street
        Houston, Texas 77002
        Attention:  Donna Lowry
        Telecopier:  (713) 646-4039

     and, if to the Company:

        Brigham Exploration Company
        6300 Bride Point Parkway
        Building 2, Suite 500
        Austin, Texas 78730
        Attention: Curtis F. Harrell
        Telecopier: (512) 472-3400

     10. GOVERNING LAW. This Warrant Certificate shall be governed by and
construed in accordance with the laws of the State of Texas without regard to
principles of conflict of laws.

                                      -11-
<PAGE>   12

     11. LOST, STOLEN, MUTILATED OR DESTROYED WARRANT CERTIFICATES. Upon receipt
by the Company of evidence reasonably satisfactory to it of the ownership of and
the loss, theft, destruction or mutilation of any Warrant Certificate, then, in
the absence of notice to the Company that such Warrant Certificate has been
acquired by a bona fide purchaser, the Company shall execute and deliver, in
exchange for or in lieu of the lost, stolen, destroyed or mutilated Warrant
Certificate, a substitute Warrant Certificate of the same tenor and evidencing a
like number of Warrants.

     12. ASSIGNMENT. The holder of this Warrant Certificate shall be entitled,
without obtaining the consent of the Company, to transfer or assign its rights,
title and interest in (and rights, title and interest under) this Warrant
Certificate in whole or in part to any Person or Persons. Upon surrender of this
Warrant Certificate to the Company, with the Transfer Form annexed hereto as
Exhibit B duly executed, the Company shall, without charge, execute and deliver
a new warrant certificate or warrant certificates, identical in form to this
Warrant Certificate, evidencing the number of Warrants being transferred
pursuant to the Transfer Form in the name of the assignee or assignees named in
such Transfer Form. If the holder's entire interest is not being assigned, the
Company shall, without charge, execute and deliver one or more new warrant
certificates identical in form to this Warrant Certificate, but evidencing a
number of Warrants equal to the number of Warrants originally represented by
this Warrant Certificate less the number being transferred pursuant to the
Transfer Form, and this Warrant Certificate shall promptly be canceled. The
terms and provisions of this Warrant Certificate shall inure to the benefit of
the holder and its successors and assigns and shall be binding upon the Company
and its successors and assigns, including, without limitation, any Person
succeeding the Company by merger, consolidation or acquisition of all or
substantially all of the Company's assets.

     13. FAIR MARKET VALUE. Whenever the terms of this Warrant require a
determination of the fair market value of the Warrants or the Common Stock and,
at the time of such determination, the Common Stock or the Warrants, as the case
may be, are not Publicly Traded, such determination shall be made in good faith
by the Board of Directors of the Company within 30 days and the Company shall
notify the holder of such determination no later than two business days
following the date of such determination. If the holder disagrees with the fair
market value as so determined by the Board of Directors, such holder shall so
notify the Company within five business days after receipt of notice from the
Board of Directors of its determination and such holder shall include in such
notice (a "Dissenting Notice"), the holder's estimate of the fair market value.
In the event the holder and the Company are unable to reach agreement regarding
the fair market value with five business days after receipt by the Company of
the Dissenting Notice, then the fair market value shall be determined by an
independent third party, knowledgeable and experienced in the valuation of
private businesses, and who is mutually acceptable to the Company and the holder
(the "Appraiser"). In the event the Company and the holder are unable to agree
upon an Appraiser within two business days after receipt by the Company of the
Dissenting Notice, each party shall, within 2 business days, select an
investment banking firm and the two investment banking firms so selected shall,
within 5 business days, select a third investment banking firm to be the
Appraiser. The Appraiser shall make a

                                      -12-
<PAGE>   13

determination of fair market value within 20 business days of his appointment
and shall notify the Company and the holder of such determination, which shall
be binding on both parties. The costs associated with the determination of fair
market value shall be borne by the Company (as such value is determined by this
Section 15, the "Fair Market Value").

                                      -13-
<PAGE>   14

         IN WITNESS WHEREOF, the Company has caused this Warrant Certificate to
be executed as of February 17, 2000, by the undersigned, thereunto duly
authorized.

                                           BRIGHAM EXPLORATION COMPANY

                                           By:
                                              ---------------------------------
                                               Curtis F. Harrell
                                               Chief Financial Officer

                                      -14-
<PAGE>   15

                                   EXHIBIT A-1

                              ELECTION TO EXERCISE

     [To be executed on exercise of the Warrants evidenced by this Warrant
     Certificate pursuant to Section 1(a)]

TO:      Brigham Exploration Company

         The undersigned, the holder of the Warrants evidenced by the attached
Warrant Certificate, hereby irrevocably elects to exercise ______________ of
such Warrants, and herewith makes payment of ___________________________
($___________) representing the aggregate Exercise Price thereof, and requests
that the certificate representing the securities issuable hereunder be issued in
the name of _____________________ and delivered to, whose address is
__________________________________________.

The Exercise Price is being paid in the following manner: [ ] by bank draft or
cashier's check in the amount of ______________________________ ($_______), [ ]
by application of ________________________ ($___________) of the outstanding
principal balance of the holder's Notes, and/or [ ] by application of
__________________________ ($__________) of interest accrued on the holder's
Notes.

     Dated:
          -------------------

                               Name of Registered Holder:
                                                         ---------------------
                               Signature:
                                         -------------------------------------
                               Title:
                                     -----------------------------------------
                               Address:
                                       ----------------------------------------

NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.

                                      -15-

<PAGE>   16

                                   EXHIBIT A-2

                        ALTERNATIVE ELECTION TO EXERCISE
[To be executed on exercise of the Warrants evidenced by this Warrant
Certificate pursuant to Section 1(b)]

TO:      Brigham Exploration Company (the "Company")

         The undersigned holder of the Warrants evidenced by the attached
Warrant Certificate, hereby elects to exercise ____________ Warrants.

[ ]      The undersigned elects to receive shares of Common Stock of the Company
         in an amount equal to the Appreciation Amount (as defined in the
         Warrant), in accordance with Section 1(b) of the Warrant, and requests
         that the securities issuable hereunder be issued in the name of
         ___________, whose address is _________________________________.

     Dated:
          -------------------

                               Name of Registered Holder:
                                                         ---------------------
                               Signature:
                                         -------------------------------------
                               Title:
                                     -----------------------------------------
                               Address:
                                       ----------------------------------------

NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.

                                      -16-
<PAGE>   17

                                   EXHIBIT B

TRANSFER FORM [To be executed only upon transfer of the Warrants evidenced by
this Warrant Certificate]

         FOR VALUE RECEIVED, the undersigned hereby sells, assigns and transfers
unto ___________________________________ _________________ of the Warrants
represented by the within Warrant Certificate, together with all right, title
and interest therein, and does hereby irrevocably constitute and appoint
_____________________________________ Attorney-in-Fact, to transfer same on the
books of the Company with full power of substitution in the premises.

     Dated:
          -------------------

                               Name of Registered Holder:
                                                         ---------------------
                               Signature:
                                         -------------------------------------
                               Title:
                                     -----------------------------------------
                               Address:
                                       ----------------------------------------

WITNESS:

---------------------------------

NOTICE: The above signature(s) must correspond with the name as written on the
face of the Warrant Certificate in every detail, without alteration or
enlargement or any change whatsoever.

                                      -17-

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