Document:

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                                                                    EXHIBIT 10.6

                           EQUITY CONVERSION AGREEMENT

         This Equity Conversion Agreement (the "Agreement") dated as of February
17, 2000 (the "Closing Date") is entered into by and among Brigham Oil & Gas,
L.P., a Delaware limited partnership (the "Borrower"), Brigham Exploration
Company, a Delaware corporation (the "Issuer") and Shell Capital Inc., a
Delaware corporation and its successors and assigns (the "Convertible Lender").

                                     RECITAL

         WHEREAS, the Borrower and Bank of Montreal, as Agent for the Lenders
have entered into that certain Amended and Restated Credit Agreement of even
date herewith (the "Credit Agreement"), pursuant to the terms and conditions of
which the Convertible Lender has agreed to make certain revolving loans to the
Borrower with such loans to be evidenced by that certain note of even date
therewith made by the Borrower payable to the order of the Convertible Lender in
the aggregate principal amount of $40,000,000 (the "Convertible Note"); and

         WHEREAS, certain sums outstanding on the Convertible Note are to be
convertible into the common stock of the Issuer pursuant to certain terms and
provisions as fully set forth in this Agreement; and

         WHEREAS, the Issuer is the parent company of the Borrower and will
directly and indirectly benefit from the loans and extension of credit by the
Convertible Lender to the Borrower under the Credit Agreement and in
consideration therefore as agreed to enter into this Equity Conversion
Agreement.

         NOW, THEREFORE, known all men by these presents that the Borrower, the
Convertible Lender and the Issuer do hereby agree as follows:

                                    ARTICLE I

                               CERTAIN DEFINITIONS

         Section 1.01 Terms Defined Above. As used in this Agreement, the terms
Borrower, Convertible Lender, Convertible Note, Credit Agreement and Issuer
shall have the meanings indicated above. Capitalized terms not otherwise defined
herein shall have the meanings set forth in the Credit Agreement.

         Section 1.02 Certain Defined Terms. As used herein, the following terms
shall have the following meanings (all terms defined in this Article I, the
Credit Agreement or in other provisions of this Agreement in the singular shall
have the same meanings when used in the plural and vice versa):

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         "Additional Shares of Common Stock" shall mean all shares of Common
Stock issued by the Issuer after the Closing Date, other than Common Stock
issued pursuant to this Agreement or the Warrant Agreement.

         "Appraised Value" shall mean, in respect of any share of Common Stock
on any date herein specified, the fair saleable value of such share of Common
Stock (determined without giving effect to the discount for (i) a minority
interest or (ii) any lack of liquidity of the Common Stock or to the fact that
the Issuer may have no class of equity registered under the Exchange Act) as of
the last day of the most recent fiscal month to end prior to such date
specified, based on the value of the Issuer, as determined by an investment
banking firm (approved by the Convertible Lender and the Issuer) in accordance
with such firm's customary practices, divided by the number of outstanding
shares of Common Stock, after giving pro forma effect to the exercise or
conversion of all exercisable or Convertible Securities (including the Warrants)
for Common Stock and the payment of the exercise or conversion price therefor.

         "Board" means the Board of Directors of the Issuer.

         "Book Value" shall mean, in respect of any share of Common Stock on any
date herein specified, the consolidated book value of the Issuer as of the last
day of any month immediately preceding such date, divided by the number of
Outstanding shares of Common Stock, after giving pro forma effect to the
exercise or conversion of all exercisable or Convertible Securities (including
the Warrants) for Common Stock and the payment of the exercise or conversion
price therefor, as determined in accordance with GAAP by any firm of independent
certified public accountants of recognized national standing selected by the
Issuer and reasonably acceptable to the Convertible Lender.

         "Business Day" shall mean each day which is not a day on which banks in
Houston, Texas are generally authorized or obligated by law or executive order
to close.

         "Closing Price" shall mean with respect to a particular security, at
the end of a given day, the last reported sale price regular way or, in case no
such reported sale takes place on such day, the average of the last closing bid
and asked prices regular way, in either case on the principal national
securities exchange on which the applicable security is listed or admitted to
trading, or if not listed or admitted to trading on any national securities
exchange, (i) the closing sale price for such day reported by the NASDAQ Stock
Market if such security is traded over-the-counter and quoted in the NASDAQ
Stock Market, or (ii) if such security is so traded, but not so quoted, the
average of the closing reported bid and asked prices of such security as
reported by the NASDAQ Stock Market or any comparable system.

         "Common Stock" shall mean the common stock, $0.01 par value per share,
of the Issuer, as constituted on the Closing Date, and any capital stock into
which such Common Stock may thereafter be changed, and shall also include (i)
capital stock of the Issuer of any other class (regardless of how denominated)
issued to the holders of shares of Common Stock upon any reclassification
thereof

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which is also not preferred as to dividends or assets over any other class of
stock of the Issuer and which is not subject to redemption and (ii) shares of
common stock of any successor or acquiring corporation (as defined in Section
2.04(h) of this Agreement) received by or distributed to the holders of Common
Stock of the Issuer in the circumstances contemplated by Section 2.04(h) of this
Agreement.

         "Conversion Date" shall mean the date set out in the Conversion Notice.

         "Conversion Notice" shall mean a written notice from the Convertible
Lender to the Issuer with a copy to the Borrower as provided in Section 2.01(h)
hereof.

         "Conversion Price" shall mean the applicable per share prices as set
out in subsection 2.01 (f).

         "Convertible Amount" shall mean the dollar amount of Convertible Loans
outstanding at any time under the Convertible Note.

         "Convertible Loan Threshold Amount" shall mean at any time the greater
of (i) $45,000,000 less the proceeds from asset sales made by the Issuer used to
permanently repay Loans less the amount of any SCI Loans previously converted to
Common Stock pursuant to the terms hereof and (ii) the Borrowing Base or Target
Asset Value as determined by the Convertible Lender from time to time.

         "Convertible Loans" shall mean those portions of the SCI Loans which
are convertible into Common Stock pursuant to Section 2.01 hereof.

         "Convertible Securities" shall mean evidences of indebtedness, shares
of stock or other securities which are convertible into or exchangeable, with or
without payment of additional consideration in cash or property, for Additional
Shares of Common Stock, either immediately or upon the occurrence of a specified
date or a specified event.

         "Convertible Tranches" shall mean Tranche One, Tranche Two and Tranche
Three as described in Section 2.01(f) hereof.

         "Current Market Price" shall mean, in respect of any share of Common
Stock on any date herein specified, (a) if there shall then be a public market
for the Common Stock, the average Price per share for the 20 trading days
preceding such date; or (b) at any time that there is no public market for the
Common Stock, the fair market value per share of Common Stock on such date as
determined reasonably and in good faith by the board of directors of the Issuer
(determined without giving effect to any discount for a minority interest, any
restrictions on transferability or any lack of liquidity of the Common Stock or
to the fact that the Issuer has no class of equity registered under the Exchange
Act), such fair market value to be determined by reference to the cash price
that would be paid between a fully informed buyer and seller under no compulsion
to buy or sell, provided that (i) if Current Market Price is being determined in
connection with an issuance of shares of Common Stock,

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solely to one or more Affiliates of the Issuer, then if so requested by the
Convertible Lender, Current Market Price shall be the Appraised Value; and (ii)
Current Market Price shall never be less than Book Value.

         "Outstanding" shall mean, when used with reference to Common Stock, at
any date as of which the number of shares thereof is to be determined, all
issued shares of Common Stock, except shares then owned or held by or for the
account of the Issuer or any Subsidiary, and shall include all shares issuable
in respect of outstanding scrip or any certificates representing fractional
interests in shares of Common Stock.

         "Permitted Issuances" shall mean (i) the issuance of shares of Common
Stock upon exercise of the Warrants or pursuant to this Agreement or the
issuance of Warrants pursuant to this Agreement, (ii) the issuance of shares
relating to any benefit plan, stock option plan or any other compensation plan
offered solely to the Issuer's officers and/or employees, (iii) the issuance of
shares of Common Stock as consideration for the purchase of any property, stock,
business or securities from any Person who is not an Affiliate of the Issuer or
any Subsidiary immediately prior to such transaction whether such shares are
issued directly by the Issuer or by a Subsidiary of the Issuer in connection
with any merger, consolidation or other business combination, (iv) if there
shall then be a public market for the Common Stock, the issuance of shares of
Common Stock upon receipt by the Issuer of no less than the Current Market Price
therefor as described in clause (a) of the definition of "Current Market Price"
and (v) if there shall then be no public market for the Common Stock, the
issuance of shares of Common Stock, warrants or Convertible Securities on terms
that are at least as favorable to the Issuer as terms that could be obtained in
an arm's length transaction with third Persons not Affiliates of the Issuer or
any Subsidiary and for consideration equal to the fair value of such shares as
determined in good faith by a majority of disinterested members of the Board of
the Issuer.

         "Person" means an individual or a corporation, partnership, trust,
incorporated or unincorporated association, limited liability company, joint
venture, joint stock company, government (or an agency or political subdivision
thereof) or other entity of any kind.

         "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. 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.

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         "SCI Loans" means that portion of the Loans allocable to the
Convertible Lender from time to time, pursuant to the Credit Agreement and SCI
Assignment.

         "Subsidiary" of a Person means (i) a corporation, a majority of whose
stock with voting power, under ordinary circumstances, to elect directors is at
the time of determination, directly or indirectly, owned by such Person or by
one or more Subsidiaries of such Person, or (ii) any other entity (other than a
corporation) in which such Person or one or more Subsidiaries of such Person,
directly or indirectly, at the date of determination thereof has at least a
majority ownership interest.

         "Warrant Agreement" shall mean that certain Warrant Agreement between
Issuer and Convertible Lender of even date herewith.

         "Warrants" shall mean one or more Warrants to be issued by the Issuer
each, in the form attached as Exhibit A to the Warrant Agreement.

                                   ARTICLE II

                                   CONVERSION

         Section 2.01 Convertible Lender Elected Conversions. Up to $30,000,000
of the SCI Loans shall constitute Convertible Loans, determined as follows:

         (a) $15,000,000 of SCI Loans shall be Convertible Loans.

         (b) An additional $5,000,000 of SCI Loans shall be Convertible Loans if
the total principal amount of the Loans outstanding exceeds the Convertible Loan
Threshold Amount plus $15,000,000.

         (c) An additional $5,000,000 of SCI Loans shall be Convertible Loans if
the total principal amount of the Loans outstanding exceeds the Convertible Loan
Threshold Amount plus $20,000,000.

         (d) An additional $5,000,000 of SCI Loans shall be Convertible Loans if
the total principal amount of the Loans outstanding exceeds the Convertible Loan
Threshold Amount plus $25,000,000.

         (e) Once SCI Loans are deemed to be Convertible Loans pursuant to
clauses (a), (b), (c) and (d) above such Convertible Loans shall be the last SCI
Loans to be repaid under the Convertible Note, other than through conversion
under this Agreement.

         (f) Convertible Amounts shall be convertible at the following
Conversion Prices:

                  (i) the first $10,000,000 in Convertible Amounts (Tranche One)
         shall be convertible at $3.90 per share;

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                  (ii) the Convertible Amounts in excess of $10,000,000 and up
         to $20,000,000 (Tranche Two) shall be convertible at $6.00 per share;
         and

                  (iii) the Convertible Amounts in excess of $20,000,000
         (Tranche Three) shall be convertible at $8.00 per share.

         (g) The Convertible Lender shall have the right to convert all or any
portion of the Convertible Amount into Common Stock at any time and from time to
time (provided, however, such portion shall not be less than $5,000,000);
provided that all of Tranche One shall be converted before any portion of
Tranche Two shall be converted, and all of Tranche Two shall be converted before
any portion of Tranche Three shall be converted.

         (h) To convert, the Convertible Lender shall deliver a Conversion
Notice not more than 60 days nor less than 15 days prior to the Conversion Date
(which must be at least 90 days after the Closing Date), which notice shall set
out the amount of the SCI Loans to be so converted, the applicable Conversion
Tranche or Tranches, Conversion Price or Prices, the number of shares of Common
Stock to be delivered upon such conversion and the Conversion Date. The number
of shares of Common Stock to be delivered upon a conversion shall be the
quotient obtained by dividing the amount of the SCI Loans (of the applicable
Conversion Tranche) to be converted by the applicable Conversion Price. If
Eurodollar Loans are outstanding and are to be repaid pursuant to any such
conversion, the Conversion Date must be a date which coincides with the last day
of the appropriate Eurodollar Interest Period.

         (i) Upon any repayment of the SCI Loans, whether at Maturity or as a
prepayment (whether or not required pursuant to Section 2.07 of the Credit
Agreement), which reduces the outstanding principal balance thereof below the
then existing Convertible Amount, Tranche One shall be deemed paid first,
Tranche Two shall be deemed paid second and Tranche Three shall be deemed paid
last, and the following shall apply:

                           (i) to the extent the provisions of clauses (ii) and
                  (iii) below are not applicable with respect to such payment,
                  the Convertible Lender's Warrants shall be exercisable in
                  accordance with the terms thereof.

                           (ii) If such repayment is made on or before the six
                  (6) month anniversary of the Closing Date, the Borrower may
                  elect to repay the outstanding principal amount of the
                  Convertible Loans so repaid at the greater of (x) 125% of par
                  or (y) $5,000,000 plus the principal amount at par less the
                  actual interest earned from the Closing Date on the
                  Convertible Amount so repaid, and in such event the conversion
                  rights with respect to the Convertible Amount so repaid shall
                  be automatically canceled.

                           (iii) From and after the six (6) month anniversary of
                  Closing Date, the Borrower may irrevocably elect by written
                  notice to the Convertible Lender, such

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                  notice given no more than 60 or no less than 30 days prior to
                  the repayment date selected by the Borrower, subject to the
                  prior conversion of any Convertible Amount by the Convertible
                  Lender, to repay all or a portion of the Convertible Amount of
                  the SCI Loans at the following premiums to par:

                           If paid between the following dates:

                           Six month anniversary to December 31, 2000       150%
                           January 1, 2001 to June 30, 2001                 140%
                           July 1, 2001 to December 31, 2001                130%
                           January 1, 2002 to June 30, 2002                 120%
                           July 1, 2002 to December 31, 2002                110%

                  Any Convertible Amount not converted by the Convertible Lender
                  prior to the repayment date elected by the Borrower shall be
                  repaid as aforesaid. If Borrower timely repays the Convertible
                  Amount so noticed on the repayment date, the conversion rights
                  with respect to the Convertible Amount so repaid shall be
                  automatically canceled.

         Section 2.02 Borrower Forced Conversion. At any time after the first
anniversary of the Closing Date, the Borrower shall have the right to require
the conversion of all or any portion of the Convertible Amount of the SCI Loans
into Common Stock upon 5 business days' notice, provided that there exists no
Event of Default nor an event which would with the passage of time result in an
Event of Default, and provided that the following additional conditions are met:

         (a) All or any portion of the Tranche One Convertible Amount may be
converted (provided, however, such portion shall not be less than $5,000,000) if
the Closing Price of the Common Stock has (i) equaled or exceeded 200% of the
Conversion Price for Tranche One on each of the five (5) Business Days preceding
the date the forced conversion election is made and (ii) averaged 200% or more
of the Conversion Price for Tranche One for the two months preceding the date
the forced conversion election is made, provided that the first day of the two
month time period shall not occur prior to the completion of 50% of the primary
wells on the CAPEX Plan approved at Closing, as such plan is amended from time
to time with the consent of the Convertible Lender.

         (b) All or any portion of the Tranche Two Convertible Amount may be
converted (provided, however, such portion shall not be less than $5,000,000) if
the Closing Price of the Common Stock has (i) equaled or exceeded 150% of the
Conversion Price for Tranche Two on each of the five (5) Business Days preceding
the date the forced conversion election is made and (ii) has averaged 150% or
more of the Conversion Price for Tranche Two for the two months preceding the
date the forced conversion election is made, provided that the first day of the
two month time period shall not occur prior to the completion of 50% of the
primary wells on the CAPEX Plan approved at Closing, as such plan is amended
from time to time with the consent of the Convertible Lender.

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         (c) All or any portion of the Tranche Three Convertible Amount may be
converted (provided, however, such portion shall not be less than $5,000,000) if
the Closing Price of the Common Stock has (i) equaled or exceeded 150% of the
Conversion Price for Tranche Three on each of the five (5) Business Days
preceding the date the forced conversion election is made and (ii) has averaged
150% or more of the Conversion Price for Tranche Three for the two months
preceding the date the forced conversion election is made, provided that the
first day of the two month time period shall not occur prior to the completion
of 50% of the primary wells on the CAPEX Plan approved at Closing, as such plan
is amended from time to time with the consent of the Convertible Lender.

         (d) All or any portion of the Convertible Amount (provided, however,
such portion shall not be less than $5,000,000) may be converted if the Issuer
conducts a public offering of Common Stock at a price of $9 or more per share of
Common Stock within twelve (12) months after the Closing Date, provided that the
Convertible Lender is permitted to sell in such public offering all or any
portion of the Common Stock acquired pursuant to such conversion on terms no
less favorable to the Convertible Lender than those contained in the
Registration Rights Agreement.

         (e) Notwithstanding anything stated in this Section 2.02 to the
contrary, the Borrower may not elect, pursuant to this Section 2.02, to convert
an amount of SCI Loans in excess of $20,000,000, unless any remaining balance of
the Convertible Note is repaid pursuant to the terms of the Credit Agreement.

         Section 2.03 Issuance of Common Stock on Conversion. The Issuer shall
deliver or cause to be delivered to the Convertible Lender on the Conversion
Date certificates representing the number of fully paid and nonassessable shares
of Common Stock into which Conversion Amount is to be converted in accordance
with the provisions hereof. Such conversion shall be deemed to have been made at
the close of business on the date the certificates are delivered and the
principal of the SCI Loans shall be deemed to be repaid by an amount equal to
such converted Conversion Amount on such date. Accrued interest on such SCI
Loans shall remain due and owing and payable at the time and as otherwise
provided in the Credit Agreement.

         No fractional shares of Common Stock shall be issued upon conversion.
Instead of any fractional shares of Common Stock which would otherwise be
issuable upon conversion, the Issuer shall pay a cash adjustment in respect of
such fraction in an amount equal to such fraction of a share multiplied by the
Conversion Price.

         Section 2.04 Adjustments and Other Rights. The Conversion Price and the
number of shares issuable upon conversion rights listed in this Agreement shall
be subject to adjustment from time to time as follows:

         (a) Stock Dividends, Subdivisions and Combinations. If at any time the
Issuer shall:

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                  (i) take a record of the holders of its Common Stock for the
         purpose of entitling them to receive a dividend payable in, or other
         distribution of, Additional Shares of Common Stock,

                  (ii) subdivide its outstanding shares of Common Stock into a
         larger number of shares of Common Stock, or

                  (iii) combine its outstanding shares of Common Stock into a
         smaller number of shares of Common Stock,

then (i) the number of shares of Common Stock for which conversion rights are
exercisable immediately after the occurrence of any such event shall be adjusted
to equal the number of shares of Common Stock which a record holder of the same
number of shares of Common Stock for which Convertible Loans are exercisable
immediately prior to the occurrence of such event would own or be entitled to
receive after the happening of such event, and (ii) the Conversion Price shall
be adjusted to equal the Conversion Price multiplied by a fraction, the
numerator of which shall be the number of shares of Common Stock for which
Convertible Loans are exercisable immediately prior to the adjustment and the
denominator of which shall be the number of shares for which Convertible Loans
are exercisable immediately after such adjustment.

         (b) Certain Other Distributions. If at any time the Issuer shall take a
record of the holders of its Common Stock for the purpose of entitling them to
receive any dividend or other distribution of:

                  (i) cash;

                  (ii) any evidences of its indebtedness (other than Convertible
         Securities), any shares of its stock (other than Additional Shares of
         Common Stock or Convertible Securities) or any other securities or
         property of any nature whatsoever (other than cash); or

                  (iii) any warrants or other rights to subscribe for or
         purchase any evidences of its indebtedness (other than Convertible
         Securities), any shares of its stock (other than Additional Shares of
         Common Stock or Convertible Securities) or any other securities or
         property of any nature whatsoever;

then (i) the number of shares of Common Stock for which Convertible Loans are
exercisable shall be adjusted to equal the product obtained by multiplying the
number of shares of Common Stock for which Convertible Loans are exercisable
immediately prior to such adjustment by a fraction (A) the numerator of which
shall be the Current Market Price per share of Common Stock at the date of
taking such record and (B) the denominator of which shall be such Current Market
Price per share of Common Stock, minus the amount allocable to one share of
Common Stock of any such cash so distributable and of the fair value (as
determined reasonably and in good faith by the board of directors of the Issuer)
of any and all such evidences of indebtedness, shares of stock, other securities

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or property or warrants or other subscription or purchase rights so
distributable, and (ii) the Conversion Price shall be adjusted to equal (A) the
Conversion Price multiplied by the number of shares of Common Stock for which
Convertible Loans are exercisable immediately prior to the adjustment divided by
(B) the number of shares for which Convertible Loans are exercisable immediately
after such adjustment. A reclassification of the Common Stock (other than a
change in par value, or from par value to no par value or from no par value to
par value) into shares of Common Stock and shares of any other class of stock
shall be deemed a distribution by the Issuer to the holders of its Common Stock
of such shares of such other class of stock within the meaning of this Section
2.04(b) and, if the outstanding shares of Common Stock shall be changed into a
larger or smaller number of shares of Common Stock as a part of such
reclassification, such change shall be deemed a subdivision or combination, as
the case may be, of the outstanding shares of Common Stock within the meaning of
Section 2.04(a) of this Agreement.

         (c) Issuance of Additional Shares of Stock.

                  (i) If at any time the Issuer shall (except as hereinafter
         provided) issue or sell any Additional Shares of Common Stock, other
         than Permitted Issuances, for consideration in an amount per Additional
         Share of Common Stock less than the Current Market Price, then the
         Conversion Price shall be adjusted by multiplying the Conversion Price
         by a fraction, the numerator of which shall be (A) an amount equal to
         the sum of (X) the number of shares of Common Stock outstanding
         immediately prior to such issuance or sale multiplied by the Current
         Market Price immediately prior to the first to occur of (i) Board
         action by the Issuer authorizing such action or (ii) the public
         announcement of an intent to take such action, plus (Y) the
         consideration, if any, received by the Issuer upon such issuance or
         sale, and the denominator of which shall be (B) the total number of
         shares of Common Stock outstanding immediately after such issuance or
         sale multiplied by the Current Market Price as determined in clause (A)
         above.

                  (ii) The provisions of Section 2.04(c)(i) of this Agreement
         shall not apply to any issuance of Additional Shares of Common Stock
         for which an adjustment is provided under Sections 2.04(a) or 2.04(b)
         of this Agreement. No adjustment of the number of shares of Common
         Stock for which Convertible Loans are exercisable shall be made under
         Section 2.04(c) of this Agreement upon the issuance of any Additional
         Shares of Common Stock which are issued pursuant to the exercise of any
         warrants or other subscription or purchase rights or pursuant to the
         exercise of any conversion or exchange rights in any Convertible
         Securities (i) if any such adjustment shall previously have been made
         upon the issuance of such warrants or other rights or upon the issuance
         of such Convertible Securities (or upon the issuance of any such
         warrants or other rights) pursuant to Section 2.04(d) or Section
         2.04(e) of this Agreement, (ii) if no adjustment was required pursuant
         to such sections upon the issuance of such Convertible Securities,
         warrants or other rights or (iii) in the event the issuance of such
         Convertible Securities, warrants or other rights predates or is of the
         same date as this Agreement, if no adjustment would have been required
         pursuant to such sections upon such issuance had this Agreement been in
         effect.

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         (d) Issuance of Warrants or Other Rights. If at any time the Issuer
shall take a record of the holders of its Common Stock for the purpose of
entitling them to receive a distribution of, or shall in any manner (whether
directly or by assumption in a merger in which the Issuer is the surviving
corporation) issue or sell, any warrants or other rights to subscribe for or
purchase any Additional Shares of Common Stock or any Convertible Securities,
other than Permitted Issuances, whether or not the rights to exchange or convert
thereunder are immediately exercisable, and if the price per share for which
Common Stock is issuable upon the exercise of such warrants or other rights or
upon conversion or exchange of such Convertible Securities shall be less than
the Current Market Price in effect immediately prior to the time of such
distribution, issue or sale, then the Conversion Price shall be adjusted as
provided in Section 2.04(a) of this Agreement on the basis that (A) the maximum
number of Additional Shares of Common Stock issuable pursuant to all such
warrants or other rights or necessary to effect the conversion or exchange of
all such Convertible Securities shall be deemed to be outstanding immediately
following such issuance, (B) the price per share for such Additional Shares of
Common Stock shall be deemed to be the lowest possible price per share in any
range of prices per share at which such Additional Shares of Common Stock are
available to such holders, and (C) the Issuer shall be deemed to have received
all of the consideration payable therefor, if any, as of the date of the actual
issuance of such warrants or other rights. No further adjustments of the
Conversion Price shall be made upon the actual issuance of such Common Stock or
of such other rights or upon exercise of such warrants or other rights or upon
the actual issuance of such Common Stock upon such conversion or exchange of
such Convertible Securities.

         (e) Issuance of Convertible Securities. If at any time the Issuer shall
take a record of the holders of its Common Stock for the purpose of entitling
them to receive a distribution of, or shall in any manner (whether directly or
by assumption in a merger in which the Issuer is the surviving corporation)
issue or sell, any Convertible Securities, other than Permitted Issuances,
whether or not the rights to exchange or convert thereunder are immediately
exercisable, and if the price per share for which Common Stock is issuable upon
such conversion or exchange shall be less than the Current Market Price in
effect immediately prior to the time of such issue or sale of Convertible
Securities, then the Conversion Price shall be adjusted as provided in Section
2.04(a) of this Agreement on the basis that (A) the maximum number of Additional
Shares of Common Stock necessary to effect the conversion or exchange of all
such Convertible Securities shall be deemed to be outstanding immediately
following such issuance, (B) the price per share of such Additional Shares of
Common Stock shall be deemed to be the lowest possible price in any range of
prices at which such Additional Shares of Common Stock are available to such
holders, and (C) the Issuer shall be deemed to have received all of the
consideration payable therefor, if any, as of the date of actual issuance of
such Convertible Securities. No adjustment of the Conversion Price shall be made
under this Section 2.04(e) upon the issuance of any Convertible Securities which
are issued pursuant to the exercise of any warrants or other subscription or
purchase rights therefor if any such adjustments shall previously have been made
upon the issuance of such warrants or other rights pursuant to Section 2.04(d)
of this Agreement. No further adjustments of the Conversion Price shall be made
upon the actual issue of such Common Stock upon conversion or exchange of such
Convertible Securities and, if any issue or sale of such Convertible Securities
is made upon exercise of any warrant or other right to purchase any such
Convertible Securities for which adjustments of the Conversion Price have been
or are to

                                       11
<PAGE>   12

be made pursuant to other provisions of this Section 2.04, no further
adjustments of the Conversion Price shall be made by reason of such issue or
sale.

         (f) Superseding Adjustment. If, at any time after any adjustment of the
Conversion Price shall have been made pursuant to Section 2.04(d) or Section
2.04(e) of this Agreement as the result of any issuance of warrants, options,
rights or Convertible Securities, and such warrants, options or rights, or the
right of conversion or exchange in such other Convertible Securities, shall
expire, and all or a portion of such warrants, options or rights, or the right
of conversion or exchange with respect to all or a portion of such other
Convertible Securities, as the case may be, shall not have been exercised, then
such previous adjustment shall be rescinded and annulled in whole or in part, as
applicable, and, if applicable, the Conversion Price shall be recalculated as if
all such expired and unexercised warrants, options, rights or Convertible
Securities had never been issued.

         (g) Other Provisions Applicable to Adjustments Under This Section. The
following provisions shall be applicable to the making of adjustments of the
number of shares of Common Stock for which Convertible Loans are exercisable
provided for in this Section 2.04:

                  (i) Computation of Consideration. To the extent that any
         Additional Shares of Common Stock shall be issued for cash
         consideration, the consideration received by the Issuer therefor shall
         be the amount of the cash received by the Issuer therefor, or, if such
         Additional Shares of Common Stock are sold to underwriters or dealers
         for public offering without a subscription offering, the initial public
         offering price (in any such case subtracting any amounts paid or
         receivable for accrued interest or accrued dividends, but not
         subtracting any compensation, discounts or expenses paid or incurred by
         the Issuer for and in the underwriting of, or otherwise in connection
         with, the issuance thereof). To the extent that such issuance shall be
         for a consideration other than cash, then, except as herein otherwise
         expressly provided, the amount of such consideration shall be deemed to
         be the fair value of such consideration at the time of such issuance as
         determined reasonably and in good faith by a majority of the
         disinterested members of the Board of the Issuer.

                  (ii) When Adjustments to Be Made. The adjustments required by
         this Section 2.04 shall be made whenever and as often as any specified
         event requiring an adjustment shall occur, except that any adjustment
         to the number of shares for which Convertible Loans are exercisable
         that would otherwise be required may be postponed (except in the case
         of a subdivision or combination of shares of the Common Stock, as
         provided for in Section 2.04(a) of this Agreement) up to, but not
         beyond, the date and time of exercise of conversion rights if such
         adjustment either by itself or with other adjustments not previously
         made adds or subtracts less than 1% to the number of shares of Common
         Stock for which conversion rights are exercisable immediately prior to
         the making of such adjustment. Any adjustment representing a change of
         less than such minimum amount (except as aforesaid) which is postponed
         shall be carried forward and made as soon as such adjustment, together
         with other adjustments required by this Section 2.04 and not previously
         made, would result in a minimum adjustment or on the date of exercise.
         For the purpose of any adjustment, any

                                       12
<PAGE>   13

         specified event shall be deemed to have occurred at the close of
         business on the date of its occurrence.

                  (iii) Fractional Interests. In computing adjustments under
         this Section 2.04, fractional interests in Common Stock resulting from
         an issuance of additional Common Stock to Convertible Lender pursuant
         to this Section 4 shall be taken into account to the nearest 1/10th of
         a share.

                  (iv) When Adjustment Not Required. If the Issuer shall take a
         record of the holders of its Common Stock for the purpose of entitling
         them to receive a dividend or distribution or subscription or purchase
         rights and shall, thereafter and before the distribution to
         stockholders thereof, legally abandon its plan to pay or deliver such
         dividend, distribution, subscription or purchase rights, no adjustment
         shall be required by reason of the taking of such record and any such
         adjustment previously made in respect thereof shall be rescinded and
         annulled.

                  (v) Escrow of Warrant Stock. If after any property becomes
         distributable pursuant to this Section 2.04 by reason of the taking of
         any record of the holders of Common Stock, but prior to the occurrence
         of the event for which such record is taken, Convertible Lender
         exercises its conversion rights, any Additional Shares of Common Stock
         issuable upon exercise of such conversion rights by reason of such
         adjustment shall be held in escrow for Convertible Lender by the Issuer
         to be issued to Convertible Lender upon and to the extent that the
         event actually takes place, upon payment of the balance, if any, of the
         Conversion Price for such Common Stock at such date (after taking into
         account any overpayment of the Conversion Price made at any time of the
         initial exercise of conversion rights). Notwithstanding any other
         provision to the contrary herein, if the event for which such record
         was taken fails to occur or is rescinded, then such escrowed shares
         shall be canceled by the Issuer and escrowed property returned.

         (h) Reorganization, Reclassification, Merger, Consolidation or
Disposition of Assets. In the event the Issuer shall reorganize its capital,
reclassify its capital stock, consolidate or merge with and into another
corporation or entity (where the Issuer is not the surviving corporation or
where there is a change in or distribution with respect to the Common Stock of
the Issuer), or sell, transfer or otherwise dispose of all or substantially all
its property, assets or business to another corporation or entity and, pursuant
to the terms of such reorganization, reclassification, merger, consolidation or
disposition of assets, shares of common stock of the successor or acquiring
corporation or entity, or any cash, shares of stock or other securities or
property of any nature whatsoever (including warrants or other subscription or
purchase rights) in addition to or in lieu of common stock of the successor or
acquiring corporation or entity ("Other Property"), are to be received by or
distributed to the holders of Common Stock of the Issuer, then the Issuer shall,
as a condition precedent to such transaction, cause effective provisions to be
made so that the Convertible Lender shall have the right thereafter to receive,
upon exercise of its conversion rights, solely the number of shares of "common
stock of the successor or acquiring corporation" or of the Issuer, if it is the
surviving corporation, and

                                       13
<PAGE>   14

Other Property receivable upon or as a result of such reorganization,
reclassification, merger, consolidation or disposition of assets, by a holder of
the number of shares of Common Stock for which Convertible Loans are exercisable
immediately prior to such event. In case of any such reorganization,
reclassification, merger, consolidation or disposition of assets, such
provisions shall include the express assumption by the successor or acquiring
corporation or entity (if other than the Issuer) of the due and punctual
observance and performance of each and every covenant and condition of this
Agreement to be performed and observed by the Issuer and all the obligations and
liabilities hereunder, subject to such modifications as may be deemed
appropriate (as determined by resolution of the board of directors of the
Issuer) to provide for adjustments of shares of the Common Stock for which
Convertible Loans are exercisable which shall be as nearly equivalent as
practicable to the adjustments provided for in this Section 2.04. For purposes
of this Section 2.04(h), "common stock of the successor or acquiring
corporation" shall include stock of such corporation of any class which is not
preferred as to dividends or assets over any other class of stock or other
securities of such corporation or entity and which is not subject to redemption
and shall also include any evidences of indebtedness, shares of stock or other
securities which are convertible into or exchangeable for any such stock or
other securities, either immediately or upon the arrival of a specified date or
the happening of a specified event, and any warrants or other rights to
subscribe for or purchase any such stock or securities. The foregoing provisions
of this Section 2.04(h) shall similarly apply to successive reorganizations,
reclassifications, mergers, consolidations or disposition of assets.

         (i) Notice of Adjustments. Whenever the number of shares of Common
Stock for which Convertible Loans are exercisable, or whenever the price at
which a share of such Common Stock may be purchased upon exercise of conversion
rights, shall be adjusted pursuant to Section 2.04, the Issuer shall forthwith
prepare a certificate to be executed by the chief financial officer of the
Issuer setting forth, in reasonable detail, the event requiring the adjustment
and the method by which such adjustment was calculated (including a description
of the basis on which the board of directors of the Issuer determined the fair
value of any evidences of indebtedness, shares of stock, other securities or
property or warrants or other subscription or purchase rights referred to in
Section 2.04 of this Agreement), specifying the number of shares of Common Stock
for which conversion rights are exercisable and (if such adjustment was made
pursuant to Section 2.04(h) of this Agreement) describing the number and kind of
any other shares of stock or Other Property for which conversion rights are
exercisable, and any change in the purchase price or prices thereof, after
giving effect to such adjustment or change. The Issuer shall promptly cause a
signed copy of such certificate to be delivered to the Convertible Lender. The
Issuer shall keep at its office or agency copies of all such certificates and
cause the same to be available for inspection at said office during normal
business hours by the Convertible Lender.

         (j) Notice of Certain Corporate Action. The Convertible Lender shall be
entitled to the same rights to receive notice of corporate action as any holder
of Common Stock.

         (k) No Impairment. The Issuer shall not by any action including,
without limitation, amending its certificate of incorporation or through any
reorganization, transfer of assets, consolidation, merger, dissolution, issue or
sale of securities or any other voluntary action, avoid or

                                       14
<PAGE>   15

seek to avoid the observance or performance of any of the terms of this
Agreement, but will at all times in good faith assist in the carrying out of all
such terms and in the taking of all such actions as may be necessary or
appropriate to protect the rights of the Convertible Lender against impairment.
Without limiting the generality of the foregoing, the Issuer will use reasonable
good faith efforts to obtain all such authorizations, exemptions or consents
from any public regulatory body having jurisdiction thereof as may be necessary
to enable it to perform its obligations under this Agreement.

         Upon the request of the Convertible Lender, the Issuer will, at any
time during the period this Agreement is in effect, acknowledge in writing, in
form satisfactory to the Convertible Lender, the continuing validity of this
Agreement and the obligations of the Issuer hereunder.

         Section 2.05 Reservation of Shares. The Issuer covenants that it will
at all times reserve and keep available, free from preemptive rights, out of its
authorized but unissued shares of Common Stock, solely for the purpose of issue
upon conversion of the Convertible Loans as herein provided, such number of
shares of Common Stock as shall then be issuable upon the conversion of the full
potential Conversion Amount. The Issuer covenants that all shares of Common
Stock which shall be so issuable shall, upon issuance, be duly and validly
issued and fully paid and non-assessable. The Issuer shall from time to time, in
accordance with applicable law, increase the authorized amount of its Common
Stock if at any time the authorized amount of shares of Common Stock remaining
unissued shall not be sufficient to permit the conversion of the full potential
Conversion Amount.

         Section 2.06 Certain Covenants. Before taking any action which would
cause an adjustment reducing the Conversion Price below the then stated or par
value of the Common Stock issuable upon conversion, the Issuer will take any
corporate action which may, in the opinion of its counsel, be necessary in order
that the Issuer may validly and legally issue fully paid and non-assessable
shares of such Common Stock at such adjusted conversion price.

         Section 2.07 Certain Notices. In case:

                  (i) the Issuer shall authorize the distribution to all holders
         of Common Stock of evidences of its indebtedness or assets (other than
         cash dividends or other cash distributions paid out of surplus); or

                  (ii) The Issuer shall authorize the granting to the holders of
         Common Stock of rights or warrants to subscribe for or purchase any
         shares of capital stock or any class or of any other rights; or

                  (iii) of any reclassification of the capital stock of the
         Issuer (other than a subdivision or combination of its outstanding
         shares of Common Stock), or of any consolidation or merger to which the
         Issuer is a party and for which approval of any stockholders of the
         Issuer is required, or of the sale, lease, or transfer of all or
         substantially all of the property of the Issuer; or

                                       15
<PAGE>   16

                  (iv) of the voluntary or involuntary dissolution, liquidation,
         or winding up of the Issuer;

then, in each case, the Issuer shall provide to the Convertible Lender at least
20 days, but not more than 45 days, prior to the applicable record or effective
date hereinafter specified, a notice stating (x) the date on which a record is
to be taken for the purpose of such dividend, distribution, rights, or warrants,
or, if a record is not to be taken, the date as of which the holders of Common
Stock of record to be entitled to such dividend, distribution, rights, or
warrants are to be determined, or (y) the date on which such reclassification,
consolidation, merger, sale, lease, transfer, dissolution, liquidation, or
winding up is expected to become effective, and the date as of which it is
expected that holders of Common Stock of record shall be entitled to exchange
their shares of Common Stock for securities or other property deliverable upon
such reclassification, consolidation, merger, sale, lease, transfer,
dissolution, liquidation, or winding up.

         Section 2.08 Board of Directors.

                  (a) The Convertible Lender shall have the right to name one
         member of the Board of the Issuer, during such time that the
         Convertible Lender owns 10% or more of the Common Stock of the Issuer.

                  (b) The Convertible Lender shall, as an observer only, have
         the right to send a representative, and such representative shall have
         a right to attend, as an observer only, all Board meetings of the
         Issuer, during such time that the Convertible Lender holds Warrants or
         Convertible Loans convertible into 10% or more of the Common Stock of
         the Issuer, and the Issuer shall give the same prior notice to the
         Convertible Lender of all Board Meetings as is given to the other Board
         members.

                                   ARTICLE III

                                  MISCELLANEOUS

         Section 3.01 Governing Law. This Agreement shall be governed by and
construed in accordance with the laws of the State of Texas.

         Section 3.02 Headings. The headings contained in this Agreement are for
reference purposes only and shall not affect in any way the meaning or
interpretation of this Agreement.

         Section 3.03 No Third Party Beneficiaries. Except as expressly provided
herein nothing in this Agreement shall entitle any person other than the parties
signatory hereto or their respective successors and assigns permitted hereby to
any claim, cause of action, remedy or right of any kind.

         Section 3.04 Successors and Assigns. All of the terms, covenants,
representations, warranties, and conditions of this Agreement shall be binding
on, and inure to the benefit of, and be

                                       16
<PAGE>   17

enforceable by the parties hereto and their respective permitted successors and
assigns. This Agreement and the rights and obligations hereunder shall be
transferred automatically from the Convertible Lender to any Person to whom the
Convertible Note, any Warrants or any Registrable Securities (as such term is
defined in the Registration Rights Agreement) are transferred, provided that (x)
the Issuer is given written notice of the transfer and the name, address,
telephone number and facsimile number of the transferee, (y) the subsequent
transferee agrees in writing to be bound by all the terms of this Agreement and
(z) at the Issuer's option, the Issuer is provided an opinion of counsel,
reasonably acceptable to the Issuer and the fees of which the Issuer shall pay,
that such transfer does not require registration under the Securities Act of
1933, as amended.

         Section 3.05 Entire Agreement. This Agreement, the Registration Rights
Agreement, the Credit Agreement, the Loan Documents and the Warrant Agreement
are intended by the parties hereto to be a complete and exclusive statement of
the agreement and understanding of the parties in respect to the subject matter
contained herein. These agreements supersede all prior agreements or
understandings among the parties with respect to such subject matter.

         Section 3.06 Binding Effect. This Agreement shall be binding upon and
shall inure to the benefit of the parties signatory hereto and their respective
successors and assigns.

         Section 3.07 Counterparts. This Agreement may be executed in any number
of counterparts, no one of which needs to be executed by all the parties, and
this Agreement shall be binding upon all the parties with the same force and
effect as if all the parties had signed the same document, and each such signed
counterpart shall constitute an original of this Agreement.

         Section 3.08 Amendments. This Agreement may be amended and the
observance of any term of this Warrant may be waived only with the written
consent of all the parties hereto.

         Section 3.09 Notice. All notices hereunder shall be in writing and
shall be effective (a) on the day on which delivered if delivered personally or
transmitted by telex or telegram or telecopier with evidence of receipt, (b) one
Business Day after the date on which the same is delivered to a nationally
recognized overnight courier service with evidence of receipt, or (c) five
Business Days after the date on which the same is deposited, postage prepaid, in
the U.S. mail, sent by certified or registered mail, return receipt requested,
and addressed to the party to be notified at the address indicated in the Credit
Agreement for the Issuer, or at the address for the Convertible Lender set forth
in the registry maintained by the Issuer or at such other address and/or
telecopy or telex number and/or to the attention of such other person as the
Issuer or the Convertible Lender may designate by ten-day advance written
notice.

                                       17
<PAGE>   18

         EXECUTED as of the date first written above.

                                                  BRIGHAM EXPLORATION COMPANY

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

                                                  BRIGHAM OIL & GAS L.P.

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

                                                  SHELL CAPITAL INC.

                                                  By:
                                                       -------------------------
                                                  Name:  Robert L. Roberts
                                                  Title: Vice-President

                                       18<PAGE>   1
                                                                    EXHIBIT 10.7

THIS WARRANT AGREEMENT AND THE SECURITIES ISSUABLE UPON EXERCISE OF THE WARRANTS
HEREIN HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED, OR
THE SECURITIES LAWS OF ANY STATE AND MAY NOT BE TRANSFERRED IN VIOLATION OF SUCH
ACT, THE RULES AND REGULATIONS PROMULGATED THEREUNDER, ANY SUCH STATE SECURITIES
LAWS OR THE PROVISIONS OF THIS WARRANT AGREEMENT.

                                WARRANT AGREEMENT

                                     for the

                            Purchase of Common Stock

                                 By and Between

                           BRIGHAM EXPLORATION COMPANY

                                       and

                               SHELL CAPITAL INC.

                                   Dated as of

                                February 17, 2000

<PAGE>   2

                                TABLE OF CONTENTS

<TABLE>
<CAPTION>
                                                                                                              Page

<S>      <C>                                                                                                  <C>
1.       DEFINITIONS.............................................................................................1

2.       ISSUANCE AND EXERCISE OF WARRANTS.......................................................................6

         2.1      Issuance of Warrants...........................................................................6

         2.2      Manner of Exercise.............................................................................6

         2.3      Payment of Taxes...............................................................................7

         2.4      Fractional Shares..............................................................................7

         2.5      Continued Validity.............................................................................8

         2.6      Conditions to Exercise.........................................................................8

3.       TRANSFERS, DIVISION AND COMBINATION.....................................................................8

         3.1      Transfer.......................................................................................8

         3.2      Division and Combination.......................................................................9

         3.3      Expenses.......................................................................................9

         3.4      Maintenance of Books...........................................................................9

4.       ADJUSTMENTS.............................................................................................9

         4.1      Stock Dividends, Subdivisions and Combinations.................................................9

         4.2      Certain Other Distributions...................................................................10

         4.3      Issuance of Additional Shares of Stock........................................................11

         4.4      Issuance of Warrants or Other Rights..........................................................11

         4.5      Issuance of Convertible Securities............................................................12

         4.6      Superseding Adjustment........................................................................13

         4.7      Other Provisions Applicable to Adjustments Under This Section.................................13

         4.8      Reorganization, Reclassification, Merger, Consolidation or Disposition
                  of Assets.....................................................................................14

5.       NOTICES TO WARRANT HOLDERS.............................................................................15

         5.1      Notice of Adjustments.........................................................................15

         5.2      Notice of Certain Corporate Action............................................................15

6.       REPRESENTATIONS AND WARRANTIES.........................................................................16

7.       CERTAIN COVENANTS......................................................................................17

         7.1      No Impairment.................................................................................17

         7.2      Reservation and Authorization of Common Stock; Registration with, or
                  Approval of, any Governmental Authority.......................................................18
</TABLE>

                                       i

<PAGE>   3

<TABLE>
<S>      <C>                                                                                                  <C>
8.       TAKING OF RECORD; STOCK AND WARRANT TRANSFER BOOKS.....................................................18

9.       RESTRICTIONS ON TRANSFERABILITY........................................................................18

         9.1      Restrictive Legend............................................................................18

         9.2      Notice of Proposed Transfers; Requests for Registration.......................................19

         9.3      Incidental Registration.......................................................................19

         9.4      Registration Procedures.......................................................................20

         9.5      Expenses......................................................................................21

         9.6      Indemnification and Contribution..............................................................21

         9.7      Termination of Restrictions...................................................................24

         9.8      Listing on Securities Exchange................................................................24

10.      SUPPLYING INFORMATION..................................................................................25

11.      LOSS OR MUTILATION.....................................................................................25

12.      OFFICE OF THE ISSUER...................................................................................25

13.      APPRAISAL..............................................................................................25

14.      LIMITATION OF LIABILITY; NO RIGHTS AS STOCKHOLDER......................................................25

15.      MISCELLANEOUS..........................................................................................26

         15.1     Non-waiver and Expenses.......................................................................26

         15.2     Notice Generally..............................................................................26

         15.3     Indemnification...............................................................................27

         15.4     Remedies......................................................................................27

         15.5     Successors and Assigns........................................................................27

         15.6     Complete Agreement; Amendment.................................................................27

         15.7     Severability..................................................................................28

         15.8     Headings......................................................................................28

         15.9     Governing Law; Consent to Jurisdiction and Venue..............................................28

         15.10    Consent to Jurisdiction and Venue.............................................................28

         15.11    Counterparts:.................................................................................29

Exhibits:

Exhibit A  -Form of Warrant Certificate...................................................................Exh. A-1

SCHEDULES:

Schedule A   -Capital Stock of the Issuer, Including Shares Subject to
               Outstanding Warrants, Options, Conversion Rights, Etc........................................Sch. A-1
</TABLE>

                                       ii

<PAGE>   4

THIS WARRANT AGREEMENT AND THE SECURITIES ISSUABLE UPON EXERCISE OF THE WARRANTS
HEREIN HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED, OR
THE SECURITIES LAWS OF ANY STATE AND MAY NOT BE TRANSFERRED IN VIOLATION OF SUCH
ACT, THE RULES AND REGULATIONS PROMULGATED THEREUNDER, ANY SUCH STATE SECURITIES
LAWS OR THE PROVISIONS OF THIS WARRANT AGREEMENT.

                                WARRANT AGREEMENT

THIS WARRANT AGREEMENT, dated as of February 17, 2000 (this "Agreement"), is
entered into by and between Brigham Exploration Company, a Delaware corporation
("Issuer"), and Shell Capital Inc. a Delaware corporation (the "Warrant Holder"
or "SCI").

                              W I T N E S S E T H:

                  WHEREAS, Brigham Oil & Gas, L.P., a limited partnership formed
under the laws of the State of Delaware (the "Borrower"), the financial
institutions party to the Credit Agreement referred to below (each a "Lender"
and collectively, the "Lenders"), and the Bank of Montreal, as agent for Lenders
under the Credit Agreement (in such capacity, the "Agent"), are parties to that
certain Amended and Restated Credit Agreement, of even date herewith (as so
amended and restated, the "Credit Agreement"); and

                  WHEREAS, the Issuer has guaranteed the obligations of the
Borrower to the Lenders and the Agent;

                  WHEREAS, as a consequence of the contractual relationships
between the Borrower and the Lenders, the Issuer has and will continue to
receive substantial benefits from the Lenders;

                  WHEREAS, in order to induce the Lenders to enter into the
Credit Agreement, the Issuer has agreed to execute and deliver this Agreement
and to issue to SCI the warrants herein described;

                  NOW, THEREFORE, in consideration of the premises and mutual
covenants contained herein, the receipt and sufficiency of which is hereby
acknowledged, the parties hereby stipulate and agree as follows:

1.       DEFINITIONS

         Capitalized terms not otherwise defined herein shall have the meaning
         set forth in the Credit Agreement. As used in this Agreement, the
         following terms have the respective meanings set forth below:

         "Additional Shares of Common Stock" shall mean all shares of Common
         Stock issued by the Issuer after the Closing Date, other than Warrant
         Stock or Common Stock issued pursuant to the Equity Conversion
         Agreement.

                                       1
<PAGE>   5

         "Appraised Value" shall mean, in respect of any share of Common Stock
         on any date herein specified, the fair saleable value of such share of
         Common Stock (determined without giving effect to the discount for (i)
         a minority interest or (ii) any lack of liquidity of the Common Stock
         or to the fact that the Issuer may have no class of equity registered
         under the Exchange Act) as of the last day of the most recent fiscal
         month to end prior to such date specified, based on the value of the
         Issuer, as determined by an investment banking firm (approved by SCI
         and the Issuer) in accordance with such firm's customary practices,
         divided by the number of Outstanding shares of Common Stock, after
         giving pro forma effect to the exercise or conversion of all
         exercisable or Convertible Securities (including the Warrants) for
         Common Stock and the payment of the exercise or conversion price
         therefor.

         "Book Value" shall mean, in respect of any share of Common Stock on any
         date herein specified, the consolidated book value of the Issuer as of
         the last day of any month immediately preceding such date, divided by
         the number of Outstanding shares of Common Stock, after giving pro
         forma effect to the exercise or conversion of all exercisable or
         Convertible Securities (including the Warrants) for Common Stock and
         the payment of the exercise or conversion price therefor, as determined
         in accordance with GAAP by any firm of independent certified public
         accountants of recognized national standing selected by the Issuer and
         reasonably acceptable to SCI.

         "Business Day" shall mean each day which is not a day on which banks in
         Houston, Texas are generally authorized or obligated by law or
         executive order to close.

         "Closing Date" shall mean the date hereof.

         "Commission" shall mean the Securities and Exchange Commission, or any
         other federal agency then administering the Securities Act and other
         federal securities laws.

         "Common Stock" shall mean the common stock, $0.01 par value per share,
         of the Issuer, as constituted on the Closing Date, and any capital
         stock into which such Common Stock may thereafter be changed, and shall
         also include (i) capital stock of the Issuer of any other class
         (regardless of how denominated) issued to the holders of shares of
         Common Stock upon any reclassification thereof which is also not
         preferred as to dividends or assets over any other class of stock of
         the Issuer and which is not subject to redemption and (ii) shares of
         common stock of any successor or acquiring corporation (as defined in
         Section 4.8 of this Agreement) received by or distributed to the
         holders of Common Stock of the Issuer in the circumstances contemplated
         by Section 4.8 of this Agreement.

         "Convertible Securities" shall mean evidences of indebtedness, shares
         of stock or other securities which are convertible into or
         exchangeable, with or without payment of additional consideration in
         cash or property, for Additional Shares of Common Stock, either
         immediately or upon the occurrence of a specified date or a specified
         event.

         "Current Market Price" shall mean, in respect of any share of Common
         Stock on any date herein specified, (a) if there shall then be a public
         market for the Common Stock, the average Price per share for the 20
         trading days preceding such date; or (b) at any time

                                       2
<PAGE>   6

         that there is no public market for the Common Stock, the fair market
         value per share of Common Stock on such date as determined reasonably
         and in good faith by the board of directors of the Issuer (determined
         without giving effect to any discount for a minority interest, any
         restrictions on transferability or any lack of liquidity of the Common
         Stock or to the fact that the Issuer has no class of equity registered
         under the Exchange Act), such fair market value to be determined by
         reference to the cash price that would be paid between a fully informed
         buyer and seller under no compulsion to buy or sell, provided that (i)
         if Current Market Price is being determined in connection with an
         issuance of shares of Common Stock, solely to one or more Affiliates of
         the Issuer, then if so requested by the Required Holders, Current
         Market Price shall be the Appraised Value; and (ii) Current Market
         Price shall never be less than Book Value.

         "Current Warrant Price" shall mean, in respect of a share of Common
         Stock at any date herein specified, three dollars ninety cents ($3.90)
         per share as to the Tranche One Warrants, six dollars ($6.00) per share
         as to the Tranche Two Warrants, and eight dollars ($8.00) per share as
         to the Tranche Three Warrants, each subject to adjustment from time to
         time as provided in this Agreement.

         "Demanding Security Holder" shall have the meaning set forth in Section
         9.3.

         "Equity Conversion Agreement" shall mean that certain Equity Conversion
         Agreement of even date herewith among the Issuer, the Borrower and SCI.

         "Exchange Act" shall mean the Securities Exchange Act of 1934, as
         amended, or any similar federal statute, and the rules and regulations
         of the Commission thereunder, all as the same shall be in effect from
         time to time.

         "Exercise Period" shall mean the period during which the Warrants are
         exercisable pursuant to Section 2.2.

         "Expiration Date" shall mean the earlier of December 30, 2003 [364 days
         after final maturity] or two years after the date of full repayment of
         the Note (as such term is defined in the Credit Agreement).

         "GAAP" shall mean generally accepted accounting principles in the
         United States of America, as from time to time in effect.

         "Material Adverse Effect" shall mean, as to the Issuer, any material
         adverse effect on the business, assets, operations, prospects or
         financial or other condition of the Issuer and its Subsidiaries, taken
         as a whole.

         "NASD" shall mean the National Association of Securities Dealers, Inc.,
         or any successor thereto.

         "Other Property" shall have the meaning set forth in Section 4.8.

         "Outstanding" shall mean, when used with reference to Common Stock, at
         any date as of which the number of shares thereof is to be determined,
         all issued shares of Common

                                       3
<PAGE>   7

         Stock, except shares then owned or held by or for the account of the
         Issuer or any Subsidiary, and shall include all shares issuable in
         respect of outstanding scrip or any certificates representing
         fractional interests in shares of Common Stock.

         "Permitted Issuances" shall mean (i) the issuance of shares of Common
         Stock upon exercise of the Warrants or pursuant to the Equity
         Conversion Agreement or the issuance of Warrants pursuant to the Equity
         Conversion Agreement, (ii) the issuance of shares relating to any
         benefit plan, stock option plan or any other compensation plan offered
         solely to the Issuer's officers and/or employees, (iii) the issuance of
         shares of Common Stock as consideration for the purchase of any
         property, stock, business or securities from any Person who is not an
         Affiliate of the Issuer or any Subsidiary immediately prior to such
         transaction whether such shares are issued directly by the Issuer or by
         a Subsidiary of the Issuer in connection with any merger, consolidation
         or other business combination, (iv) if there shall then be a public
         market for the Common Stock, the issuance of shares of Common Stock
         upon receipt by the Issuer of no less than the Current Market Price
         therefor as described in clause (a) of the definition of "Current
         Market Price" and (v) if there shall then be no public market for the
         Common Stock, the issuance of shares of Common Stock, warrants or
         Convertible Securities on terms that are at least as favorable to the
         Issuer as terms that could be obtained in an arm's length transaction
         with third Persons not Affiliates of the Issuer or any Subsidiary and
         for consideration equal to the fair value of such shares as determined
         in good faith by a majority of disinterested members of the board of
         directors of the Issuer.

         "Person(s)" shall mean any individual, sole proprietorship,
         partnership, joint venture, trust, limited liability company,
         incorporated organization, association, corporation, institution,
         public benefit corporation, entity or government (whether federal,
         state, county, city, municipal or otherwise, including, without
         limitation, any instrumentality, division, agency, body or department
         thereof).

         "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. 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.

         "Registrable Securities" shall mean, at any particular time and as to
         each Warrant Holder, (i) all shares of common stock issuable upon the
         exercise of such Warrant Holder's Warrants, (ii) all of such Warrant
         Holder's issued and outstanding Warrant Stock, and

                                       4
<PAGE>   8

         (iii) all shares of common stock issuable upon exercise of SCI's rights
         under the Equity Conversion Agreement.

         "Registration Expenses" shall have the meaning set forth in Section 9.5
         of this Agreement.

         "Registration Statement" shall have the meaning set forth in Section
         9.4 of this Agreement.

         "Required Holders" shall mean the Warrant Holders of Warrants
         exercisable for an amount exceeding 50% of the aggregate number of
         shares of Common Stock then purchasable upon exercise of all Warrants,
         whether or not exercisable.

         "Requirement of Law" shall mean, as to any Person, any requirement
         contained in any certificate of incorporation, bylaws, or other
         organizational or governing documents of such Person, and any law,
         treaty, rule or regulation or determination of an arbitrator or a court
         or other governmental authority, in each case applicable to or binding
         such Person or any of the property or to which such Person or any of
         its property is subject.

         "Restricted Common Stock" shall mean shares of Common Stock which are,
         or which upon their issuance on the exercise of a Warrant would be,
         evidenced by a certificate bearing the restrictive legend set forth in
         Section 9.1 of this Agreement.

         "Securities Act" shall mean the Securities Act of 1933, as amended, or
         any similar federal statute, and the rules and regulations of the
         Commission thereunder, all as the same shall be in effect at the time.

         "Subsidiary" of a Person means (i) a corporation, a majority of whose
         stock with voting power, under ordinary circumstances, to elect
         directors is at the time of determination, directly or indirectly,
         owned by such Person or by one or more Subsidiaries of such Person, or
         (ii) any other entity (other than a corporation) in which such Person
         or one or more Subsidiaries of such Person, directly or indirectly, at
         the date of determination thereof has at least a majority ownership
         interest.

         "Tranche One Warrants" shall have the meaning set forth in Section 2.1
         of this Agreement.

         "Tranche Two Warrants" shall have the meaning set forth in Section 2.1
         of this Agreement.

         "Tranche Three Warrants" shall have the meaning set forth in Section
         2.1 of this Agreement.

         "Transfer Notice" shall have the meaning set forth in Section 9.2 of
         this Agreement.

         "Warrant Certificate" shall mean a certificate evidencing one or more
         Warrants, substantially in the form of Exhibit A hereto, with such
         changes therein as may be required to reflect any adjustments made
         pursuant to Section 4 of this Agreement.

                                       5
<PAGE>   9

         "Warrant Holder" shall mean such Person in whose name the Warrants are
         registered on the books of the Issuer maintained for such purpose or
         each Person holding any Warrant Stock. As of the Closing Date, the SCI
         is the Warrant Holder hereof.

         "Warrant Price" shall mean, for any exercise of Warrants pursuant to
         Section 2.2 of this Agreement, an amount equal to (i) the number of
         shares of Common Stock being purchased upon such exercise multiplied by
         (ii) the Current Warrant Price applicable to Tranche One Warrants,
         Tranche Two Warrants or Tranche Three Warrants, as applicable, for each
         share of Common Stock as of the date of such exercise.

         "Warrant Stock" shall mean the shares of Common Stock purchased by the
         Warrant Holders upon the exercise thereof.

         "Warrants" shall mean the Warrants issued pursuant to this Agreement,
         including the Tranche One Warrants, Tranche Two Warrants and Tranche
         Three Warrants, and all Warrants issued upon transfer, division or
         combination of, or in substitution for, such Warrants. All Warrants
         shall at all times be identical as to terms and conditions and date,
         except as to the number of shares of Common Stock for which they may be
         exercised and the Current Warrant Price. A Warrant shall entitle the
         record holder thereof to purchase from the Issuer one share of Common
         Stock (subject to adjustment as provided in Section 4 of this
         Agreement).

2.       ISSUANCE AND EXERCISE OF WARRANTS

2.1      Issuance of Warrants. The Issuer hereby agrees to issue to the Warrant
         Holder on the Closing Date TWO MILLION FIVE HUNDRED SIXTY-FOUR THOUSAND
         ONE HUNDRED TWO (2,564,102) Warrants having a Current Warrant Price of
         three dollars and ninety cents ($3.90) per share of Common Stock (the
         "Tranche One Warrants"), ONE MILLION SIX HUNDRED SIXTY-SIX THOUSAND SIX
         HUNDRED SIXTY-SEVEN (1,666,667) Warrants having a Current Warrant Price
         of six dollars ($6.00) per share of Common Stock (the "Tranche Two
         Warrants"), and ONE MILLION TWO HUNDRED FIFTY THOUSAND (1,250,000)
         Warrants having a Current Price of eight dollars ($8.00) per share
         ("Tranche Three Warrants"). On the Closing Date, the Issuer shall
         deliver to the Warrant Holder Warrant Certificates evidencing the
         Warrants issued to the Warrant Holder.

2.2      Manner of Exercise.

         (a)      The Warrant Holder may, from and after the Closing Date until
                  11:59 p.m., Central Standard Time on the Expiration Date,
                  exercise the Warrants evidenced by a Warrant Certificate, on
                  any Business Day, for all or part of the number of shares of
                  Common Stock purchasable thereunder.

         (b)      In order to exercise the Warrants, in whole or in part, the
                  Warrant Holder shall deliver to the Issuer at its principal
                  office at 6300 Bridge Point Parkway, Building 2, Suite 500,
                  Austin, Texas 78730, Attention: Chief Financial Officer, or at
                  the office or agency designated by the Issuer pursuant to
                  Section 12 of this Agreement (the "Principal Office"), (x) a
                  written notice duly executed by the

                                       6
<PAGE>   10

                  Warrant Holder or its agent or attorney, substantially in the
                  form of the form of election to purchase appearing at the end
                  of the Warrant Certificate as Exhibit A thereto, of such
                  Warrant Holder's election to exercise the Warrants, which
                  notice shall specify the number of shares of Common Stock to
                  be purchased, (y) payment of the Warrant Price in the manner
                  provided below, and (z) the Warrant Certificate or Warrant
                  Certificates evidencing the Warrants. Upon receipt thereof,
                  the Issuer shall, as promptly as practicable, and in any event
                  within three (3) Business Days thereafter, execute or cause to
                  be executed and deliver or cause to be delivered to such
                  Warrant Holder a certificate or certificates representing the
                  aggregate number of full shares of Common Stock issuable upon
                  such exercise, together with cash in lieu of any fraction of a
                  share, as hereinafter provided. The stock certificate or
                  certificates so delivered shall be, to the extent possible, in
                  such denomination or denominations as such Warrant Holder
                  shall request in the notice and shall be registered in the
                  name of the Warrant Holder or, subject to Section 9 of this
                  Agreement, such other name as shall be designated in the
                  notice. The Warrants shall be deemed to have been exercised
                  and such certificate or certificates shall be deemed to have
                  been issued, and such Warrant Holder or any other Person so
                  designated to be named therein shall be deemed to have become
                  a holder of record of such shares for all purposes, as of the
                  date the notice, together with payment of the Warrant Price
                  and the Warrant Certificate or Warrant Certificates, are
                  received by the Issuer as described above and all taxes
                  required to be paid by such Warrant Holder, if any, pursuant
                  to Section 2.3 of this Agreement prior to the issuance of such
                  shares have been paid. If the Warrants evidenced by a Warrant
                  Certificate shall have been exercised, the Issuer shall, at
                  the time of delivery of the certificate or certificates
                  representing the Warrant Stock, deliver to the Warrant Holder
                  a new Warrant Certificate evidencing the rights of the Warrant
                  Holder to purchase the unpurchased shares of Common Stock
                  represented by the old Warrant Certificate, which new Warrant
                  Certificate shall in all other respects be identical to the
                  old Warrant Certificate. Payment of the Warrant Price shall be
                  made in cash in an amount equal to the Warrant Price.

2.3      Payment of Taxes. The Issuer shall pay all expenses in connection with,
         and all transfer taxes and other governmental charges that may be
         imposed with respect to, the issuance or delivery of Warrant Stock. The
         Issuer shall not be required, however, to pay any tax or other charge
         imposed in connection with any transfer involved in the issue of any
         certificate for shares of Warrant Stock issuable upon exercise of
         Warrants in any name other than that of Warrant Holder, and in such
         case the Issuer shall not be required to issue or deliver any stock
         certificate until such tax or other charge has been paid or it has been
         established to the satisfaction of the Issuer that no such tax or other
         charge is due.

2.4      Fractional Shares. The Issuer shall not be required to issue a
         fractional share of Common Stock upon the exercise of Warrants as
         provided in Section 2(b)(i) and (ii). As to any fraction of a share
         which the Warrant Holder would otherwise be entitled to purchase upon
         such exercise, the Issuer shall pay a cash adjustment in respect of
         such fraction in an amount equal to the same fraction of the Current
         Market Price per share of Common Stock on the date of exercise. If the
         determination of Current Market Price for purposes of this Section 2.4
         would otherwise require an appraisal to be made by an investment

                                       7
<PAGE>   11

         banking firm, then Current Market Price for purposes of this Section
         2.4 only shall mean Book Value per share of Common Stock on the date of
         exercise, unless a determination of Appraised Value shall have been
         made within six months prior to such date in which case such Appraised
         Value shall be utilized for the purposes of determining Current Market
         Price.

2.5      Continued Validity.

         A holder of Warrant Stock (other than a holder who acquires such shares
         after the same have been publicly sold pursuant to a Registration
         Statement under the Securities Act) shall continue to be entitled with
         respect to such shares to all rights to which it would have been
         entitled as a holder of Warrant Stock under Sections 9, 10 and 15 of
         this Agreement. The Issuer will, at the time of each exercise of
         Warrants or upon the request of the holder of Warrant Stock issued upon
         the exercise thereof, acknowledge in writing, in form reasonably
         satisfactory to such holder of Warrant Stock, its continuing obligation
         to afford to such holder of Warrant Stock all such rights; provided,
         however, that if such holder of Warrant Stock shall fail to make any
         such request, such failure shall not affect the continuing obligation
         of the Issuer to afford to such holder of Warrant Stock all such
         rights.

2.6      Conditions to Exercise.

         Notwithstanding any other provision of this Agreement, the Warrants
         shall be exercisable only (i) upon any repayment of the SCI Loans,
         whether at Maturity or as a prepayment (whether or not required
         pursuant to Section 2.07 of the Credit Agreement), but only to the
         extent that (A) such repayment reduces the outstanding principal
         balance thereof below the then existing Convertible Amount, and (B) the
         provisions of clauses (ii) and (iii) of Section 2.01(i) of the Equity
         Conversion Agreement are not applicable with respect to such repayment.
         The Warrants shall be exercisable in the following order: the Tranche
         One Warrants shall become exercisable first, as the SCI Loans
         constituting the Tranche One loans are repaid; the Tranche Two Warrants
         shall become exercisable second, as the SCI Loans constituting the
         Tranche Two loans are repaid; and Tranche Three Warrants shall become
         exercisable last, as the SCI Loans constituting the Tranche Three loans
         are repaid. Further, notwithstanding any other provision of this
         Agreement, the number of shares purchasable pursuant to such Warrants
         shall be the quotient of that part of the Convertible Amount of the
         relevant Tranche so repaid divided by the relevant Current Warrant
         Price.

3.       TRANSFERS, DIVISION AND COMBINATION

3.1      Transfer. Subject to compliance with Section 9 of this Agreement,
         transfer of Warrants, in whole or in part, shall be registered on the
         books of the Issuer to be maintained for such purposes, upon surrender
         of the Warrant Certificate representing such Warrants at the principal
         office of the Issuer referred to in Section 2.2 of this Agreement or
         the office or agency designated by the Issuer pursuant to Section 12 of
         this Agreement, together with a written assignment substantially in the
         form of Exhibit B to the Warrant Certificate and a written agreement,
         in form reasonably satisfactory to the Issuer, setting forth the

                                       8
<PAGE>   12

         new Warrant Holder's agreement to be bound by all of the terms of this
         Agreement each duly executed by the Warrant Holder or its agent or
         attorney, and funds sufficient to pay any transfer taxes payable by
         such Warrant Holder upon the making of such transfer. Upon such
         surrender and, if required, such payment, the Issuer shall, subject to
         Section 9 of this Agreement, execute and deliver a new Warrant
         Certificate or Warrant Certificates in the name of the assignee or
         assignees and in the denomination specified in such instrument of
         assignment, and shall issue to the assignor a new Warrant Certificate
         or Warrant Certificates evidencing the portion of the old Warrant
         Certificate not so assigned, and the old Warrant Certificate shall
         promptly be canceled. A Warrant, if properly assigned in compliance
         with Section 9 of this Agreement, may be exercised by a new Warrant
         Holder for the purchase of shares of Warrant Stock without having a new
         Warrant Certificate or new Warrant Certificates issued.

3.2      Division and Combination. Subject to the provisions of Section 9 of
         this Agreement, any Warrant Certificate may be divided or combined with
         other Warrant Certificates upon presentation thereof at the aforesaid
         office or agency of the Issuer, together with a written notice
         specifying the names and denominations in which new Warrant
         Certificates are to be issued, signed by a Warrant Holder or its agent
         or attorney. Subject to compliance with Section 3.1 of this Agreement
         as to any transfer which may be involved in such division or
         combination, the Issuer shall execute and deliver a new Warrant
         Certificate or Warrant Certificates in exchange for the Warrant
         Certificate or Warrant Certificates to be divided or combined in
         accordance with such notice.

3.3      Expenses. The Issuer shall prepare, issue and deliver at its own
         expense (other than transfer taxes) the new Warrant Certificate or
         Warrant Certificates provided for under this Section 3.

3.4      Maintenance of Books. The Issuer agrees to maintain, at its aforesaid
         office or agency, books for the registration of, and the registration
         of transfer of, the Warrants.

4.       ADJUSTMENTS

         The number of shares of Warrant Stock for which Warrants are
         exercisable, and the price at which such shares may be purchased upon
         exercise of Warrants, shall be subject to adjustment from time to time
         as set forth in this Section 4. The Issuer shall give each Warrant
         Holder notice of any event described below which requires an adjustment
         pursuant to this Section 4 within three (3) Business Days after such
         event.

4.1      Stock Dividends, Subdivisions and Combinations. If at any time the
         Issuer shall:

         (a)      take a record of the holders of its Common Stock for the
                  other distribution of, Additional Shares of Common Stock,

         (b)      subdivide its outstanding shares of Common Stock into a larger
                  number of shares of Common Stock, or

                                       9
<PAGE>   13

         (c)      combine its outstanding shares of Common Stock into a smaller
                  number of shares of Common Stock,

                  then (i) the number of shares of Common Stock for which a
                  Warrant is exercisable immediately after the occurrence of any
                  such event shall be adjusted to equal the number of shares of
                  Common Stock which a record holder of the same number of
                  shares of Common Stock for which a Warrant is exercisable
                  immediately prior to the occurrence of such event would own or
                  be entitled to receive after the happening of such event, and
                  (ii) the Current Warrant Price shall be adjusted to equal the
                  Current Warrant Price multiplied by a fraction, the numerator
                  of which shall be the number of shares of Common Stock for
                  which a Warrant is exercisable immediately prior to the
                  adjustment and the denominator of which shall be the number of
                  shares for which a Warrant is exercisable immediately after
                  such adjustment.

4.2      Certain Other Distributions. If at any time the Issuer shall take a
         record of the holders of its Common Stock for the purpose of entitling
         them to receive any dividend or other distribution of:

         (a)      cash;

         (b)      any evidences of its indebtedness (other than Convertible
                  Securities), any shares of its stock (other than Additional
                  Shares of Common Stock or Convertible Securities) or any other
                  securities or property of any nature whatsoever (other than
                  cash); or

         (c)      any warrants or other rights to subscribe for or purchase any
                  evidences of its indebtedness (other than Convertible
                  Securities), any shares of its stock (other than Additional
                  Shares of Common Stock or Convertible Securities) or any other
                  securities or property of any nature whatsoever;

                  then (i) the number of shares of Common Stock for which a
                  Warrant is exercisable shall be adjusted to equal the product
                  obtained by multiplying the number of shares of Common Stock
                  for which a Warrant is exercisable immediately prior to such
                  adjustment by a fraction (A) the numerator of which shall be
                  the Current Market Price per share of Common Stock at the date
                  of taking such record and (B) the denominator of which shall
                  be such Current Market Price per share of Common Stock, minus
                  the amount allocable to one share of Common Stock of any such
                  cash so distributable and of the fair value (as determined
                  reasonably and in good faith by the board of directors of the
                  Issuer) of any and all such evidences of indebtedness, shares
                  of stock, other securities or property or warrants or other
                  subscription or purchase rights so distributable, and (ii) the
                  Current Warrant Price shall be adjusted to equal (A) the
                  Current Warrant Price multiplied by the number of shares of
                  Common Stock for which a Warrant is exercisable immediately
                  prior to the adjustment divided by (B) the number of shares
                  for which a Warrant is exercisable immediately after such
                  adjustment. A reclassification of the Common Stock (other than
                  a change in par value, or from

                                       10
<PAGE>   14

                  par value to no par value or from no par value to par value)
                  into shares of Common Stock and shares of any other class of
                  stock shall be deemed a distribution by the Issuer to the
                  holders of its Common Stock of such shares of such other class
                  of stock within the meaning of this Section 4.2 and, if the
                  Outstanding shares of Common Stock shall be changed into a
                  larger or smaller number of shares of Common Stock as a part
                  of such reclassification, such change shall be deemed a
                  subdivision or combination, as the case may be, of the
                  Outstanding shares of Common Stock within the meaning of
                  Section 4.1 of this Agreement.

4.3      Issuance of Additional Shares of Stock.

         (a)      If at any time the Issuer shall (except as hereinafter
                  provided) issue or sell any Additional Shares of Common Stock,
                  other than Permitted Issuances, for consideration in an amount
                  per Additional Share of Common Stock less than the Current
                  Market Price, then the Current Warrant Price shall be adjusted
                  by multiplying the Current Warrant Price by a fraction, the
                  numerator of which shall be (A) an amount equal to the sum of
                  (X) the number of shares of Common Stock Outstanding
                  immediately prior to such issuance or sale multiplied by the
                  Current Market Price immediately prior to the first to occur
                  of (i) board action by the Issuer authorizing such action or
                  (ii) the public announcement of an intent to take such action,
                  plus (Y) the consideration, if any, received by the Issuer
                  upon such issuance or sale, and the denominator of which shall
                  be (B) the total number of shares of Common Stock Outstanding
                  immediately after such issuance or sale multiplied by the
                  Current Market Price as determined in clause (A) above.

         (b)      The provisions of Section 4.3(a) of this Agreement shall not
                  apply to any issuance of Additional Shares of Common Stock for
                  which an adjustment is provided under Sections 4.1 or 4.2 of
                  this Agreement. No adjustment of the number of shares of
                  Common Stock for which a Warrant shall be exercisable shall be
                  made under Section 4.3(a) of this Agreement upon the issuance
                  of any Additional Shares of Common Stock which are issued
                  pursuant to the exercise of any warrants or other subscription
                  or purchase rights or pursuant to the exercise of any
                  conversion or exchange rights in any Convertible Securities
                  (i) if any such adjustment shall previously have been made
                  upon the issuance of such warrants or other rights or upon the
                  issuance of such Convertible Securities (or upon the issuance
                  of any such warrants or other rights) pursuant to Section 4.4
                  or Section 4.5 of this Agreement, (ii) if no adjustment was
                  required pursuant to such sections upon the issuance of such
                  Convertible Securities, warrants or other rights or (iii) in
                  the event the issuance of such Convertible Securities,
                  warrants or other rights predates or is of the same date as
                  this Agreement, if no adjustment would have been required
                  pursuant to such sections upon such issuance had this
                  Agreement been in effect.

4.4      Issuance of Warrants or Other Rights. If at any time the Issuer shall
         take a record of the holders of its Common Stock for the purpose of
         entitling them to receive a distribution of, or shall in any manner
         (whether directly or by assumption in a merger in which the Issuer

                                       11
<PAGE>   15

         is the surviving corporation) issue or sell, any warrants or other
         rights to subscribe for or purchase any Additional Shares of Common
         Stock or any Convertible Securities, other than Permitted Issuances,
         whether or not the rights to exchange or convert thereunder are
         immediately exercisable, and if the price per share for which Common
         Stock is issuable upon the exercise of such warrants or other rights or
         upon conversion or exchange of such Convertible Securities shall be
         less than the Current Market Price in effect immediately prior to the
         time of such distribution, issue or sale, then the Current Warrant
         Price shall be adjusted as provided in Section 4.3(a) of this Agreement
         on the basis that (A) the maximum number of Additional Shares of Common
         Stock issuable pursuant to all such warrants or other rights or
         necessary to effect the conversion or exchange of all such Convertible
         Securities shall be deemed to be Outstanding immediately following such
         issuance, (B) the price per share for such Additional Shares of Common
         Stock shall be deemed to be the lowest possible price per share in any
         range of prices per share at which such Additional Shares of Common
         Stock are available to such holders, and (C) the Issuer shall be deemed
         to have received all of the consideration payable therefor, if any, as
         of the date of the actual issuance of such warrants or other rights. No
         further adjustments of the Current Warrant Price shall be made upon the
         actual issuance of such Common Stock or of such other rights or upon
         exercise of such warrants or other rights or upon the actual issuance
         of such Common Stock upon such conversion or exchange of such
         Convertible Securities.

4.5      Issuance of Convertible Securities. If at any time the Issuer shall
         take a record of the holders of its Common Stock for the purpose of
         entitling them to receive a distribution of, or shall in any manner
         (whether directly or by assumption in a merger in which the Issuer is
         the surviving corporation) issue or sell, any Convertible Securities,
         other than Permitted Issuances, whether or not the rights to exchange
         or convert thereunder are immediately exercisable, and if the price per
         share for which Common Stock is issuable upon such conversion or
         exchange shall be less than the Current Market Price in effect
         immediately prior to the time of such issue or sale of Convertible
         Securities, then the Current Warrant Price shall be adjusted as
         provided in Section 4.3(a) of this Agreement on the basis that (A) the
         maximum number of Additional Shares of Common Stock necessary to effect
         the conversion or exchange of all such Convertible Securities shall be
         deemed to be Outstanding immediately following such issuance, (B) the
         price per share of such Additional Shares of Common Stock shall be
         deemed to be the lowest possible price in any range of prices at which
         such Additional Shares of Common Stock are available to such holders,
         and (C) the Issuer shall be deemed to have received all of the
         consideration payable therefor, if any, as of the date of actual
         issuance of such Convertible Securities. No adjustment of the Current
         Warrant Price shall be made under this Section 4.5 upon the issuance of
         any Convertible Securities which are issued pursuant to the exercise of
         any warrants or other subscription or purchase rights therefor if any
         such adjustments shall previously have been made upon the issuance of
         such warrants or other rights pursuant to Section 4.4 of this
         Agreement. No further adjustments of the Current Warrant Price shall be
         made upon the actual issue of such Common Stock upon conversion or
         exchange of such Convertible Securities and, if any issue or sale of
         such Convertible Securities is made upon exercise of any warrant or
         other right to purchase any such Convertible Securities for which
         adjustments of the Current Warrant Price have been or are to be made
         pursuant to other provisions of this Section 4,

                                       12
<PAGE>   16

         no further adjustments of the Current Warrant Price shall be made by
         reason of such issue or sale.

4.6      Superseding Adjustment. If, at any time after any adjustment of the
         Current Warrant Price shall have been made pursuant to Section 4.4 or
         Section 4.5 of this Agreement as the result of any issuance of
         warrants, options, rights or Convertible Securities, and such warrants,
         options or rights, or the right of conversion or exchange in such other
         Convertible Securities, shall expire, and all or a portion of such
         warrants, options or rights, or the right of conversion or exchange
         with respect to all or a portion of such other Convertible Securities,
         as the case may be, shall not have been exercised, then such previous
         adjustment shall be rescinded and annulled and, if applicable, the
         Current Warrant Price shall be recalculated as if all such expired and
         unexercised warrants, options, rights or Convertible Securities had
         never been issued.

4.7      Other Provisions Applicable to Adjustments Under This Section. The
         following provisions shall be applicable to the making of adjustments
         of the number of shares of Common Stock for which a Warrant is
         exercisable provided for in this Section 4:

         (a)      Computation of Consideration. To the extent that any
                  Additional Shares of Common Stock shall be issued for cash
                  consideration, the consideration received by the Issuer
                  therefor shall be the amount of the cash received by the
                  Issuer therefor, or, if such Additional Shares of Common Stock
                  are sold to underwriters or dealers for public offering
                  without a subscription offering, the initial public offering
                  price (in any such case subtracting any amounts paid or
                  receivable for accrued interest or accrued dividends, but not
                  subtracting any compensation, discounts or expenses paid or
                  incurred by the Issuer for and in the underwriting of, or
                  otherwise in connection with, the issuance thereof). To the
                  extent that such issuance shall be for a consideration other
                  than cash, then, except as herein otherwise expressly
                  provided, the amount of such consideration shall be deemed to
                  be the fair value of such consideration at the time of such
                  issuance as determined reasonably and in good faith by a
                  majority of the disinterested members of the board of
                  directors of the Issuer.

         (b)      When Adjustments to Be Made. The adjustments required by this
                  Section 4 shall be made whenever and as often as any specified
                  event requiring an adjustment shall occur, except that any
                  adjustment to the number of shares for which the Warrants are
                  exercisable that would otherwise be required may be postponed
                  (except in the case of a subdivision or combination of shares
                  of the Common Stock, as provided for in Section 4.1 of this
                  Agreement) up to, but not beyond, the date and time of
                  exercise of any Warrants if such adjustment either by itself
                  or with other adjustments not previously made adds or
                  subtracts less than 1% to the number of shares of Common Stock
                  for which the Warrants initially issued pursuant to this
                  Agreement are exercisable immediately prior to the making of
                  such adjustment. Any adjustment representing a change of less
                  than such minimum amount (except as aforesaid) which is
                  postponed shall be carried forward and made as soon as such
                  adjustment, together with other adjustments required by this
                  Section 4 and not previously made, would result in a minimum

                                       13
<PAGE>   17

                  adjustment or on the date of exercise. For the purpose of any
                  adjustment, any specified event shall be deemed to have
                  occurred at the close of business on the date of its
                  occurrence.

         (c)      Fractional Interests. In computing adjustments under this
                  Section 4, fractional interests in Common Stock resulting from
                  an issuance of additional Warrants to any Warrant Holder
                  pursuant to this Section 4 shall be taken into account to the
                  nearest 1/10th of a share, subject to Section 2.4 of this
                  Agreement.

         (d)      When Adjustment Not Required. If the Issuer shall take a
                  record of the holders of its Common Stock for the purpose of
                  entitling them to receive a dividend or distribution or
                  subscription or purchase rights and shall, thereafter and
                  before the distribution to stockholders thereof, legally
                  abandon its plan to pay or deliver such dividend,
                  distribution, subscription or purchase rights, no adjustment
                  shall be required by reason of the taking of such record and
                  any such adjustment previously made in respect thereof shall
                  be rescinded and annulled.

         (e)      Escrow of Warrant Stock. If after any property becomes
                  distributable pursuant to this Section 4 by reason of the
                  taking of any record of the holders of Common Stock, but prior
                  to the occurrence of the event for which such record is taken,
                  any Warrant Holder exercises Warrants, any Additional Shares
                  of Common Stock issuable upon exercise of such Warrant by
                  reason of such adjustment shall be held in escrow for a
                  Warrant Holder by the Issuer to be issued to such Warrant
                  Holder upon and to the extent that the event actually takes
                  place, upon payment of the balance, if any, of the Warrant
                  Price for such Warrant at such date (after taking into account
                  any overpayment of the Warrant Price made at any time of the
                  initial Warrant exercise). Notwithstanding any other provision
                  to the contrary herein, if the event for which such record was
                  taken fails to occur or is rescinded, then such escrowed
                  shares shall be canceled by the Issuer and escrowed property
                  returned.

4.8      Reorganization, Reclassification, Merger, Consolidation or Disposition
         of Assets. In the event the Issuer shall reorganize its capital,
         reclassify its capital stock, consolidate or merge with and into
         another corporation or entity (where the Issuer is not the surviving
         corporation or where there is a change in or distribution with respect
         to the Common Stock of the Issuer), or sell, transfer or otherwise
         dispose of all or substantially all its property, assets or business to
         another corporation or entity and, pursuant to the terms of such
         reorganization, reclassification, merger, consolidation or disposition
         of assets, shares of common stock of the successor or acquiring
         corporation or entity, or any cash, shares of stock or other securities
         or property of any nature whatsoever (including warrants or other
         subscription or purchase rights) in addition to or in lieu of common
         stock of the successor or acquiring corporation or entity ("Other
         Property"), are to be received by or distributed to the holders of
         Common Stock of the Issuer, then the Issuer shall, as a condition
         precedent to such transaction, cause effective provisions to be made so
         that each Warrant Holder shall have the right thereafter to receive,
         upon exercise of a warrant, solely the number of shares of "common
         stock of the successor or acquiring corporation" or of the Issuer, if
         it is the surviving corporation, and Other Property receivable upon or
         as a result of such reorganization, reclassification, merger,

                                       14
<PAGE>   18

         consolidation or disposition of assets, by a holder of the number of
         shares of Common Stock for which a Warrant is exercisable immediately
         prior to such event. In case of any such reorganization,
         reclassification, merger, consolidation or disposition of assets, such
         provisions shall include the express assumption by the successor or
         acquiring corporation or entity (if other than the Issuer) of the due
         and punctual observance and performance of each and every covenant and
         condition of this Agreement to be performed and observed by the Issuer
         and all the obligations and liabilities hereunder, subject to such
         modifications as may be deemed appropriate (as determined by resolution
         of the board of directors of the Issuer) to provide for adjustments of
         shares of the Common Stock for which a Warrant is exercisable which
         shall be as nearly equivalent as practicable to the adjustments
         provided for in this Section 4. For purposes of this Section 4.8,
         "common stock of the successor or acquiring corporation" shall include
         stock of such corporation of any class which is not preferred as to
         dividends or assets over any other class of stock or other securities
         of such corporation or entity and which is not subject to redemption
         and shall also include any evidences of indebtedness, shares of stock
         or other securities which are convertible into or exchangeable for any
         such stock or other securities, either immediately or upon the arrival
         of a specified date or the happening of a specified event, and any
         warrants or other rights to subscribe for or purchase any such stock or
         securities. The foregoing provisions of this Section 4.8 shall
         similarly apply to successive reorganizations, reclassifications,
         mergers, consolidations or disposition of assets.

5.       NOTICES TO WARRANT HOLDERS

5.1      Notice of Adjustments. Whenever the number of shares of Common Stock
         for which a Warrant is exercisable, or whenever the price at which a
         share of such Common Stock may be purchased upon exercise of the
         Warrants, shall be adjusted pursuant to Section 4, the Issuer shall
         forthwith prepare a certificate to be executed by the chief financial
         officer of the Issuer setting forth, in reasonable detail, the event
         requiring the adjustment and the method by which such adjustment was
         calculated (including a description of the basis on which the board of
         directors of the Issuer determined the fair value of any evidences of
         indebtedness, shares of stock, other securities or property or warrants
         or other subscription or purchase rights referred to in Section 4 of
         this Agreement), specifying the number of shares of Common Stock for
         which a Warrant is exercisable and (if such adjustment was made
         pursuant to Section 4.8 of this Agreement) describing the number and
         kind of any other shares of stock or Other Property for which a Warrant
         is exercisable, and any change in the purchase price or prices thereof,
         after giving effect to such adjustment or change. The Issuer shall
         promptly cause a signed copy of such certificate to be delivered to
         each Warrant Holder in accordance with Section 15.2 of this Agreement.
         The Issuer shall keep at its office or agency designated pursuant to
         Section 12 of this Agreement copies of all such certificates and cause
         the same to be available for inspection at said office during normal
         business hours by any Warrant Holder or any prospective purchaser of a
         Warrant designated by a Warrant Holder thereof.

5.2      Notice of Certain Corporate Action. Each Warrant Holder shall be
         entitled to the same rights to receive notice of corporate action as
         any holder of Common Stock.

                                       15
<PAGE>   19

6.       REPRESENTATIONS AND WARRANTIES

         The Issuer makes the following representations and warranties, each and
         all of which shall be true and correct as of the date of execution and
         delivery of this Agreement and shall survive the execution and delivery
         of this Agreement:

         (a)      Due Organization; Etc. The Issuer is a corporation duly
                  organized validly existing and in good standing under the laws
                  of the State of Delaware, and has the power and authority to
                  execute and deliver this Agreement and the Warrant
                  Certificates, to issue the Warrants and to perform its
                  obligations under this Agreement and the Warrant Certificates.

         (b)      Due Authorization; No Violation. The execution, delivery and
                  performance by the Issuer of this Agreement and the Warrant
                  Certificates, the issuance of the Warrants and the issuance of
                  the Warrant Stock upon exercise of the Warrants have been duly
                  authorized by all necessary corporate action and do not and
                  will not violate, or result in a breach of, or constitute a
                  default under or require any consent under, or result in the
                  creation of any lien or security interest upon the assets of
                  the Issuer pursuant to, any Requirement of Law or any
                  contractual obligation binding upon the Issuer.

         (c)      Due Execution; Etc. This Agreement has been duly executed and
                  delivered by the Issuer and constitutes a legal, valid and
                  enforceable obligation of the Issuer. When the Warrants and
                  the Warrant Certificates have been issued as contemplated
                  hereby, (i) the Warrants and the Warrant Certificates will
                  constitute legal, valid, binding and enforceable obligations
                  of the Issuer and (ii) the Warrant Stock, when issued upon
                  exercise of the Warrants in accordance with the terms hereof,
                  will be duly authorized, validly issued, fully paid and
                  non-assessable shares of Common Stock with no personal
                  liability attaching to the ownership thereof.

         (d)      Capitalization. The total number of shares of all classes of
                  stock that the Issuer shall on the Closing Date have authority
                  to issue is 40,000,000 shares, consisting of (i) 30,000,000
                  shares of Common Stock, par value $0.01 per share, of which,
                  after giving effect to the transactions contemplated herein
                  and all other issuances of capital stock of the Issuer on or
                  prior to the Closing Date, 14,428,621 shares of Common Stock
                  will be issued and outstanding and 5,480,769 shares of Common
                  Stock will be reserved for future issuance pursuant to this
                  Agreement and (ii) 10,000,000 shares of Preferred Stock, par
                  value $0.01 per share, none of which are presently
                  outstanding. Schedule A sets forth a complete list of the
                  outstanding capital stock of the Issuer, including any
                  options, warrants or rights to purchase the capital stock of
                  the Issuer. The delivery hereunder by the Issuer to the
                  Warrant Holder of the Warrants issued on the Closing Date will
                  transfer and convey to the Warrant Holder good and marketable
                  title to such Warrants and, upon exercise of such Warrants in
                  accordance with this Agreement, good and marketable title to
                  the Common Stock purchased upon such exercise, free and clear
                  of all preemptive rights, liens, charges and encumbrances,
                  except for

                                       16
<PAGE>   20

                  restrictions on transfer referred to in this Agreement, or
                  arising under the Federal and state securities laws. Except as
                  otherwise disclosed on Schedule A, the Issuer does not have
                  outstanding any stock or securities convertible into or
                  exchangeable for any shares of its stock, nor, except as so
                  set forth, does it have outstanding any agreements, rights or
                  options entitling any person to subscribe for or to purchase
                  any capital stock or securities convertible into or
                  exchangeable for any of its shares of stock.

         (e)      Full Disclosure. No information contained in this Agreement,
                  the financial statements referred to in the Credit Agreement
                  or any written statement furnished by or on behalf of the
                  Issuer pursuant to the terms of this Agreement to the Warrant
                  Holder contains any untrue statement of a material fact or
                  omits to state a material fact necessary to make the
                  statements contained herein or therein not misleading in light
                  of the circumstances under which made.

         (f)      Warrant Price. The Issuer has taken all corporate action, and
                  obtained all necessary authorizations or exemptions from any
                  public regulatory body or bodies or governmental entity or
                  entities having jurisdiction thereof, as may be necessary in
                  order that the Issuer may validly and legally issue fully paid
                  and non-assessable shares of Common Stock upon to exercise of
                  the warrants at the Warrant Price, as the same may be adjusted
                  pursuant hereto.

         (g)      Other Representations and Warranties. The Issuer hereby
                  affirms and reaffirms for the express benefit of the Warrant
                  Holders that the representations and warranties made by the
                  Issuer in that certain Guaranty Agreement dated as of January
                  26, 1998, as amended, are true and correct, as if made in
                  favor of the Warrant Holder on the date hereof.

7.       CERTAIN COVENANTS

7.1      No Impairment. The Issuer shall not by any action including, without
         limitation, amending its certificate of incorporation or through any
         reorganization, transfer of assets, consolidation, merger, dissolution,
         issue or sale of securities or any other voluntary action, avoid or
         seek to avoid the observance or performance of any of the terms of this
         Agreement, but will at all times in good faith assist in the carrying
         out of all such terms and in the taking of all such actions as may be
         necessary or appropriate to protect the rights of each Warrant Holder
         against impairment. Without limiting the generality of the foregoing,
         the Issuer will use reasonable good faith efforts to obtain all such
         authorizations, exemptions or consents from any public regulatory body
         having jurisdiction thereof as may be necessary to enable it to perform
         its obligations under this Agreement.

         Upon the request of a Warrant Holder, the Issuer will, at any time
         during the period this Agreement is in effect, acknowledge in writing,
         in form satisfactory to such Warrant Holder, the continuing validity of
         this Agreement and the obligations of the Issuer hereunder.

                                       17
<PAGE>   21

7.2      Reservation and Authorization of Common Stock; Registration with, or
         Approval of, any Governmental Authority. From and after the Closing
         Date, the Issuer shall at all times reserve and keep available for
         issue upon the exercise of Warrants such number of its authorized but
         unissued shares of Common Stock as will be sufficient to permit the
         exercise in full of all outstanding Warrants. All shares of Common
         Stock which shall be so issuable, when issued upon exercise of any
         Warrants and payment therefor in accordance with the terms of this
         Agreement, shall be duly and validly issued and fully paid and
         non-assessable, and not subject to preemptive rights.

         Before taking any action which would result in an adjustment in the
         number of shares of Common Stock for which a Warrant is exercisable or
         in the Current Warrant Price, the Issuer shall obtain all such
         authorizations or exemptions thereof, or consents thereto, as may be
         necessary from any public regulatory body or bodies or governmental
         entity or entities having jurisdiction thereof.

         If any shares of Common Stock required to be reserved for issuance upon
         exercise of Warrants require registration or qualification with any
         governmental authority under any federal or state law (otherwise than
         as provided in Section 9 of this Agreement) before such shares may be
         so issued, the Issuer will in good faith and as expeditiously as
         possible and at its expense endeavor to cause such shares to be duly
         registered.

8.       TAKING OF RECORD; STOCK AND WARRANT TRANSFER BOOKS

         In the case of all dividends or other distributions by the Issuer to
         the holders of its Common Stock with respect to which any provision of
         Section 4 of this Agreement refers to the taking of a record of such
         holders, the Issuer will in each such case take such a record as of the
         close of business on a Business Day. The Issuer will not at any time,
         except upon dissolution, liquidation or winding up of the Issuer, close
         its stock transfer books or Warrant transfer books so as to result in
         preventing or delaying the exercise or transfer of any Warrants.

9.       RESTRICTIONS ON TRANSFERABILITY

         The Warrants and the Warrant Stock shall not be transferred before
         satisfaction of the conditions specified in this Section 9, which
         conditions are intended to ensure compliance with the provisions of the
         Securities Act and applicable state securities laws with respect to the
         transfer of any Warrant or any Warrant Stock. Each Warrant Holder, by
         entering into this Agreement and accepting the Warrants, agrees to be
         bound by the provisions of this Section 9.

9.1      Restrictive Legend. Except as otherwise provided in this Section 9,
         each certificate representing Warrants or Warrant Stock, shall be
         stamped or otherwise imprinted with a legend in substantially the
         following form:

         "THE SECURITIES REPRESENTED BY THIS CERTIFICATE HAVE NOT BEEN
         REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED, OR ANY
         APPLICABLE STATE SECURITIES LAWS, AND MAY NOT BE SOLD OR TRANSFERRED IN
         THE ABSENCE OF SUCH REGISTRATION OR AN

                                       18
<PAGE>   22

         EXEMPTION THEREFROM. SUCH SECURITIES ARE SUBJECT TO THE RESTRICTIONS
         AND PRIVILEGES SPECIFIED IN A WARRANT AGREEMENT, DATED AS OF FEBRUARY
         17, 2000, BETWEEN BRIGHAM EXPLORATION COMPANY AND THE INITIAL HOLDERS
         OF SECURITIES NAMED THEREIN, A COPY OF WHICH IS ON FILE WITH THE
         SECRETARY OF BRIGHAM EXPLORATION COMPANY AND WILL BE FURNISHED WITHOUT
         CHARGE TO THE HOLDER HEREOF UPON WRITTEN REQUEST, AND THE HOLDER OF
         THIS CERTIFICATE AGREES TO BE BOUND THEREBY."

9.2      Notice of Proposed Transfers; Requests for Registration. Prior to any
         transfer of any Warrants or any shares of Restricted Common Stock, the
         Warrant Holder of such Warrants or Restricted Common Stock shall give
         five days prior written notice to the Issuer of such Warrant Holder's
         intention to effect such transfer (a "Transfer Notice"). Each Warrant
         Holder agrees that it will not sell, transfer or otherwise dispose of
         Warrants or any shares of Restricted Common Stock, in whole or in part,
         except pursuant to an effective registration statement under the
         Securities Act or an exemption from registration thereunder. Each
         certificate, if any, evidencing such shares of Restricted Common Stock
         issued upon such transfer shall bear the restrictive legend set forth
         in Section 9.1, and each Warrant Certificate issued upon such transfer
         shall bear the restrictive legend set forth in Section 9.1 of this
         Agreement, unless in the opinion of the transferee's or Warrant
         Holder's counsel delivered to the Issuer in connection with such
         transfer such legend is not required in order to ensure compliance with
         the Securities Act.

         The Warrant Holders of Warrants and Warrant Stock shall have the right
         to request registration of such Warrant Stock pursuant to Section 9.3
         of this Agreement.

9.3      Incidental Registration. If the Issuer at any time proposes to file on
         its behalf and/or on behalf of any of its security holders (the
         "Demanding Security Holders") a Registration Statement under the
         Securities Act on any form (other than a Registration Statement (i)
         filed pursuant to demand under the Company's Registration Rights
         Agreement with Joint Energy Development Investments II Limited
         Partnership, a Delaware limited partnership, and Enron Capital & Trade
         Resources Corp., a Delaware corporation, dated August 20, 1998, as
         amended, or (ii) on Form S-8 or any similar or successor form or any
         other registration statement relating to an offering of securities
         solely to the Issuer's existing security holders or employees) to
         register the offer and sale of its Common Stock for cash, it will give
         written notice to all Warrant Holders of Warrants or Warrant Stock at
         least twenty (20) days before the anticipated date of initial filing
         with the Commission of such Registration Statement, which notice shall
         set forth the Issuer's intention to effect such a registration, the
         class or series and number of equity securities proposed to be
         registered and the intended method of disposition of the securities
         proposed to be registered by the Issuer. The notice shall offer to
         include in such filing all of the Warrant Holder's Registrable
         Securities.

         Each Warrant Holder desiring to have Registrable Securities registered
         under this Section 9.3 shall advise the Issuer in writing within
         fifteen (15) days after the date of receipt of such offer from the
         Issuer, setting forth the amount of such Registrable

                                       19
<PAGE>   23
         Securities for which registration is requested. The Issuer shall
         thereupon include in such filing the number of shares of Registrable
         Securities for which registration is so requested, subject to the next
         sentence, and shall use its best efforts to effect registration under
         the Securities Act of such securities. If the managing underwriter of a
         proposed public offering shall advise the Issuer in writing that, in
         its opinion, the distribution of the Registrable Securities requested
         to be included in the registration concurrently with the securities
         being registered by the Issuer or any Demanding Security Holder would
         materially and adversely affect the distribution of such securities by
         the Issuer or such Demanding Security Holders, then all selling
         security holders (but not the Issuer or the Demanding Security Holders)
         shall reduce the amount of securities each intended to distribute
         through such offering on a pro rata basis to the greatest aggregate
         amount which, in the opinion of such managing underwriter, would not
         materially and adversely affect the distribution of such securities.

         Nothing in this Section 9.3 shall preclude the Issuer from
         discontinuing the registration of its securities being effected on its
         behalf under this Section 9.3 at any time prior to the effective date
         of the registration relating thereto. Notwithstanding any provision
         herein, the rights of the Warrant Holder under this Section 9.3 are
         subject to the express limitations contained in registration rights
         agreements in effect on the date hereof between the Issuer and other
         parties; provided, however, that the Issuer shall not on or after the
         date of this Agreement enter into any registration rights agreement
         with respect to its securities that conflict with the registration
         rights granted to the Warrant Holder herein.

9.4      Registration Procedures. If the Issuer is required by the provisions of
         this Section 9 to use its best efforts to effect the registration of
         any of its securities under the Securities Act, the Issuer will, as
         expeditiously as possible:

         (a)      prepare and file with the Commission a registration statement
                  with respect to such securities (a "Registration Statement")
                  and use its best efforts to cause such Registration Statement
                  to become and remain effective for the period described in
                  paragraph (b) below;

         (b)      prepare and file with the Commission such amendments and
                  supplements to such Registration Statement and the prospectus
                  used in connection therewith as may be necessary to keep such
                  Registration Statement effective and to comply with the
                  provisions of the Securities Act with respect to the sale or
                  other disposition of all securities covered by such
                  Registration Statement until the earlier of such time as all
                  of such securities have been disposed of in a public offering
                  or the expiration of 90 days;

         (c)      furnish to such selling security holders such number of copies
                  of a summary prospectus or other prospectus, including a
                  preliminary prospectus, in conformity with the requirements of
                  the Securities Act, and such other documents, as such selling
                  security holders may reasonably request;

                                       20
<PAGE>   24

         (d)      use its best efforts to register or qualify the securities
                  covered by such Registration Statement under such other
                  securities or blue sky laws of such jurisdictions within the
                  United States as each holder of such securities shall request
                  (provided, however, the Issuer shall not be obligated to
                  qualify as a foreign corporation to do business under the laws
                  of any jurisdiction in which it is not then qualified or to
                  file any general consent to service or process), and do such
                  other reasonable acts and things as may be required of it to
                  enable such holder to consummate the disposition in such
                  jurisdiction of the securities covered by such Registration
                  Statement;

         (e)      enter into customary agreements (including an underwriting
                  agreement in customary form) and take such other actions as
                  are reasonably required in order to expedite or facilitate the
                  disposition of such Registrable Securities; and

         (f)      otherwise use its best efforts to comply with all applicable
                  rules and regulations of the Commission, and make available to
                  its security holders, as soon as reasonably practicable, but
                  not later than 18 months after the effective date of the
                  Registration Statement, an earnings statement covering the
                  period of at least 12 months beginning with the first full
                  month after the effective date of such Registration Statement,
                  which earnings statements shall satisfy the provisions of
                  Section 11(a) of the Securities Act.

                  It shall be a condition precedent to the obligation of the
                  Issuer to take any action pursuant to this Section 9 in
                  respect of the securities which are to be registered at the
                  request of any Warrant Holder of Registrable Securities that
                  such Warrant Holder shall furnish to the Issuer such
                  information regarding the securities held by such Warrant
                  Holder and the intended method of disposition thereof as the
                  Issuer shall reasonably request and as shall be required in
                  connection with the action taken by the Issuer.

9.5      Expenses. All expenses incurred in complying with this Section 9,
         including, without limitation, all registration and filing fees
         (including all expenses incident to filing with the NASD), printing
         expenses, fees and disbursements of counsel for the Issuer, the
         reasonable fees and expenses of one counsel for the selling security
         holders (selected by the Person holding the plurality of the securities
         being registered), expenses of any special audits incident to or
         required by any such registration and expenses of complying with the
         securities or blue sky laws of any jurisdictions pursuant to Section
         9.4(d) of this Agreement (all of such expenses shall be collectively
         referred to herein as "Registration Expenses"), shall be paid by the
         Issuer; provided, however, the Issuer shall not be responsible for any
         discount or commission or cost reimbursement to any underwriter in
         respect of the securities sold by such Warrant Holder of Registrable
         Securities.

9.6      Indemnification and Contribution.

         (a)      In the event of any registration of any of the Registrable
                  Securities under the Securities Act pursuant to this Section
                  9, the Issuer shall indemnify and hold harmless each Warrant
                  Holder of such Registrable Securities, such Warrant

                                       21
<PAGE>   25

                  Holder's directors and officers, each Affiliate of such
                  Warrant Holder, and each other Person (including each
                  underwriter) who participated in the offering of such
                  Registrable Securities and each other Person, if any, who
                  controls such Warrant Holder or such participating Person, if
                  any, who controls such Warrant Holder or such participating
                  Person within the meaning of the Securities Act, against any
                  losses, claims, damages or liabilities, joint or several, to
                  which such Warrant Holder or any such director or officer or
                  participating Person or Affiliate or controlling Person may
                  become subject under the Securities Act or any other statute
                  or at common law, insofar as such losses, claims, damages or
                  liabilities (or actions in respect thereof) arise out of or
                  are based upon (i) any alleged untrue statement of any
                  material fact contained, on the effective date thereof, in any
                  Registration Statement under which such securities were
                  registered under the Securities Act, any preliminary
                  prospectus or final prospectus contained therein, or any
                  amendment or supplement thereto, or (ii) any alleged omission
                  to state therein a material fact required to be stated therein
                  or necessary to make the statements therein, in light of the
                  circumstances under which they were made, not misleading, and
                  shall reimburse such Warrant Holder or such director, officer
                  or participating Person or Affiliate or controlling Person for
                  any legal or any other expenses reasonably incurred by such
                  Warrant Holder or such director, officer or participating
                  Person or Affiliate or controlling Person in connection with
                  investigating or defending any such loss, claim, damage,
                  liability or action; provided, however, that the Issuer shall
                  not be liable in any such case to the extent that any such
                  loss, claim, damage or liability directly arises out of or is
                  directly based upon any alleged untrue statement or alleged
                  omission made in such Registration Statement, preliminary
                  prospectus, prospectus or amendment or supplement in reliance
                  upon and in conformity with written information furnished to
                  the Issuer by such Warrant Holder specifically for use
                  therein. Such indemnity shall remain in full force and effect
                  regardless of any investigation made by or on behalf of such
                  Warrant Holder or such director, officer or participating
                  Person or Affiliate or controlling Person, and shall survive
                  the transfer of such securities by such Warrant Holder.

         (b)      Each Warrant Holder of any Registrable Securities, by
                  acceptance thereof, agrees to indemnify and hold harmless the
                  Issuer, its directors and officers and each other Person, if
                  any, who controls the Issuer within the meaning of the
                  Securities Act against any losses, claims, damages or
                  liabilities, joint or several, to which the Issuer or any such
                  director or officer or any such Person may become subject
                  under the Securities Act or any other statute or at common
                  law, insofar as such losses, claims, damages or liabilities
                  (or actions in respect thereof) directly arise out of or are
                  directly based upon (i) information in writing provided to the
                  Issuer by such Warrant Holder of such Registrable Securities
                  contained, on the effective date thereof, in any Registration
                  Statement under which securities were registered under the
                  Securities Act at the request of such Warrant Holder, any
                  preliminary prospectus or final prospectus contained therein,
                  or any amendment or supplement thereto or (ii) that such
                  Warrant Holder's obligation under this Section 9.6(b) to
                  indemnify and hold harmless the Issuer shall in no event
                  exceed the lesser of (x) the damage attributable solely to the
                  inclusion of such written

                                       22
<PAGE>   26

                  information in such Registration Statement, preliminary
                  prospectus, final prospectus, or amendment or supplement
                  suffered by the Person or Persons whose claims gave rise to
                  such losses, claims, damages or liabilities and (y) the net
                  proceeds received by such Warrant Holder from the sale of
                  Registrable Securities giving rise to such indemnification.

         (c)      If the indemnification provided for in this Section 9 from the
                  indemnifying party is unavailable to an indemnified party
                  hereunder in respect of any losses, claims, damages,
                  liabilities or expenses referred to herein, then the
                  indemnifying party, in lieu of indemnifying such indemnified
                  party, shall contribute to the amount paid or payable by such
                  indemnified party as a result of such losses, claims, damages,
                  liabilities or expenses in such proportion as is appropriate
                  to reflect the relative fault of the indemnifying party and
                  indemnified parties in connection with the actions which
                  resulted in such losses, claims, damages, liabilities or
                  expenses, as well as any other relevant equitable
                  considerations. The relative fault of such indemnifying party
                  and indemnified parties shall be determined by reference to,
                  among other things, whether any action in question, including
                  any untrue or alleged untrue statement of a material fact or
                  omission or alleged omission to state a material fact, has
                  been made by, or related to information supplied by, such
                  indemnifying party or indemnified parties, and the parties'
                  relative intent, knowledge, access to information and
                  opportunity to correct or prevent such action. The amount paid
                  or payable by a party under this Section 9 as a result of the
                  losses, claims, damages, liabilities and expenses referred to
                  above shall be deemed to include any legal or other fees or
                  expenses reasonably incurred by such party in connection with
                  any investigation or proceeding.

                           The parties hereto agree that it would not be just
                  and equitable if contribution pursuant to this Section 9.6(c)
                  were determined by pro rata allocation or by any other method
                  of allocation which does not take account of the equitable
                  considerations referred to in the immediately preceding
                  paragraph. Notwithstanding the provisions of this subsection
                  (c), no Warrant Holder shall be required to contribute any
                  amount in excess of the total amount received by it upon the
                  sale of its securities pursuant to the Registration Statement
                  to which the losses, claims, damages, liabilities and expenses
                  referred to above relate. No Person guilty of fraudulent
                  misrepresentation (within the meaning of Section 11(f) of the
                  Securities Act) shall be entitled to contribution from any
                  Person who was not guilty of such fraudulent
                  misrepresentation. The obligations of each of the Warrant
                  Holders under this subsection (c) to contribute are several
                  and not joint.

         (d)      Conduct of Indemnification Proceedings. Any person or entity
                  entitled to indemnification hereunder shall (i) give prompt
                  written notice to the indemnifying party after the receipt by
                  the indemnified party of a written notice of the commencement
                  of any action, suit, proceeding or investigation or threat
                  thereof made in writing for which such indemnified party will
                  claim indemnification or contribution pursuant to this
                  Agreement; provided, however, that the failure of any
                  indemnified party to give notice as provided herein shall not
                  relieve the indemnifying party of its obligations under
                  Section 9.6 hereof, except to the

                                       23
<PAGE>   27

                  extent that the indemnifying party is actually prejudiced by
                  such failure to give notice, and (ii) unless in such
                  indemnified party's reasonable judgment a conflict of interest
                  may exist between such indemnified and indemnifying parties
                  with respect to such claim, permit such indemnifying party to
                  assume the defense of such claim with counsel reasonably
                  satisfactory to the indemnified party. If the indemnifying
                  party is entitled to, and does, assume the defense of such
                  claim, the indemnified party shall have the right to employ
                  separate counsel and to participate in the defense thereof,
                  but the fees and expenses of such counsel shall be borne by
                  the indemnified party. Whether or not such defense is assumed
                  by the indemnifying party, the indemnifying party shall not be
                  subject to any liability for any settlement made without its
                  consent (but such consent will not be unreasonably withheld).
                  No indemnifying party shall be permitted to consent to the
                  entry of any judgment or to enter into any settlement that
                  does not include as an unconditional term thereof the giving
                  by the claimant or plaintiff to such indemnified party of a
                  release from all liability in respect of such claim or
                  litigation. An indemnifying party who is not entitled to, or
                  elects not to, assume the defense of a claim shall not be
                  obligated to pay the fees and expenses of more than one
                  counsel in any one jurisdiction for all parties indemnified by
                  such indemnifying party with respect to such claim, unless in
                  the reasonable judgment of any indemnified party a conflict of
                  interest may exist between such indemnified party and any
                  other of such indemnified parties with respect to such claim,
                  in which event the indemnifying party shall be obligated to
                  pay the fees and expenses of such additional counsel or
                  counsels.

9.7      Termination of Restrictions. Notwithstanding the foregoing provisions
         of this Section 9, the restrictions imposed by this Section 9 upon the
         transferability of the Warrants, the Warrant Stock and the Restricted
         Common Stock (or Common Stock issuable upon the exercise of the
         Warrants) and the legend requirement of Section 9.1 of this Agreement
         shall terminate as to any particular Warrant or share of Warrant Stock
         or Restricted Common Stock (or Warrant Stock) (i) when and so long as
         such security shall have been registered under the Securities Act and
         disposed of pursuant thereto, or (ii) when the Warrant Holder thereof
         shall have delivered to the Issuer the written opinion of counsel to
         such Warrant Holder, stating that such legend is not required in order
         to ensure compliance with the Securities Act. Whenever the restrictions
         imposed by this Section 9 shall terminate as to any Warrants or any
         Restricted Common Stock, as hereinabove provided, the Warrant Holder
         thereof shall be entitled to receive from the Issuer, at the expense of
         the Issuer, a new Warrant Certificate or a new certificate representing
         such Common Stock, as the case may be, not bearing the restrictive
         legend set forth in Section 9.1 of this Agreement.

9.8      Listing on Securities Exchange. If at any time the Issuer shall list
         any shares of Common Stock on any securities exchange, it will, at its
         expense, use its best efforts to list thereon, maintain and, when
         necessary, increase such listing of, all shares of Common Stock issued
         or, to the extent permissible under the applicable securities exchange
         rules, issuable upon the exercise of the Warrants so long as any shares
         of Common Stock shall be so listed during the Exercise Period.

                                       24
<PAGE>   28

10.      SUPPLYING INFORMATION

         The Issuer shall cooperate with each Warrant Holder of a Warrant and
         each Warrant Holder of Restricted Common Stock in supplying such
         information as may be reasonably necessary for such Warrant Holder to
         complete and file any information reporting forms presently or
         hereafter required by the Commission as a condition to the availability
         of an exemption from the Securities Act for the sale of any Warrant or
         Restricted Common Stock.

11.      LOSS OR MUTILATION

         Upon receipt by the Issuer from any Warrant Holder of evidence
         reasonably satisfactory to it of the ownership of and the loss, theft,
         destruction or mutilation of a certificate representing Warrants or
         Warrant Stock and indemnity reasonably satisfactory to it (it being
         understood that the written agreement of the Warrant Holder or an
         Affiliate thereof shall be sufficient indemnity) and in case of
         mutilation upon surrender and cancellation hereof or thereof, the
         Issuer will execute and deliver in lieu hereof or thereof a new Warrant
         or new stock certificate as the case may be, of like tenor to such
         Warrant Holder; provided, in the case of mutilation, no indemnity shall
         be required if the certificate representing Warrants or Warrant Stock
         in identifiable form is surrendered to the Issuer for cancellation.

12.      OFFICE OF THE ISSUER

         As long as any of the Warrants remain outstanding, the Issuer shall
         maintain an office or agency (which may be the principal executive
         officers of the Issuer) where the Warrants may be presented for
         exercise, registration or transfer, division or combination as provided
         in this Agreement.

13.      APPRAISAL

         The determination of the Appraised Value per share of Common Stock
         shall be made by an investment banking firm of nationally recognized
         standing mutually agreed to by the Issuer and the Required Holders. If
         the investment banking firm selected by the Issuer is not acceptable to
         the Required Holders and the Issuer and the Required Holders cannot
         agree on a mutually acceptable investment banking firm, then the
         Required Holders and the Issuer shall each choose one such investment
         banking firm and the respective chosen firms shall agree on another
         investment banking firm which shall make the determination. The Issuer
         shall retain, at its sole cost, such investment banking firm as may be
         necessary for the determination of Appraised Value required by the
         terms of this Agreement.

14.      LIMITATION OF LIABILITY; NO RIGHTS AS STOCKHOLDER

         No provision hereof, in the absence of affirmative action by any
         Warrant Holder to purchase shares of Common Stock, and no enumeration
         herein of the rights or privileges of any Warrant Holder, shall give
         rise to any liability of such Warrant Holder for the purchase price of
         any Common Stock or as a stockholder of the Issuer, whether such

                                       25
<PAGE>   29

         liability is asserted by the Issuer or by creditors of the Issuer.
         Except as may otherwise be provided by law or by separate agreement
         between a Warrant Holder and the Issuer, no Warrant Holder, as such,
         shall be entitled to vote or be deemed the holder of Common Stock or
         any other securities (other than Warrants) of the Issuer which may at
         any time be issuable on the exercise hereof, nor shall anything
         contained herein be construed to confer upon any Warrant Holder the
         rights of a stockholder of the Issuer or the right to vote for the
         election of directors or upon any matters submitted to stockholders at
         any meeting thereof, or to give or withhold consent to any corporate
         action or to receive notice of meetings or other actions affecting
         stockholders (except as provided herein), or to receive dividends or
         otherwise, until the Warrants shall have been exercised in accordance
         with the terms and conditions hereof.

15.      MISCELLANEOUS

15.1     Non-waiver and Expenses. No course of dealing or any delay or failure
         to exercise any right hereunder on the part of any holder of Warrant
         Stock shall operate as a waiver of such right or otherwise prejudice
         such holder of Warrant Stock's rights, powers or remedies. If the
         Issuer fails to comply with any provision of this Agreement, the Issuer
         shall pay to the applicable holder of Warrant Stock such amounts as
         shall be sufficient to cover any costs and expenses including, but not
         limited to, reasonable attorneys' fees, including those of appellate
         proceedings, incurred by the holder of Warrant Stock in enforcing any
         of its rights, powers or remedies hereunder.

15.2     Notice Generally. Any notice, demand, request, consent, approval,
         declaration, delivery or other communication hereunder to be made
         pursuant to the provisions of this Agreement shall be sufficiently
         given or made if in writing and either delivered in person with receipt
         acknowledged or sent by registered or certified mail, return receipt
         requested, postage prepaid, telex, telecopier or overnight air courier
         guaranteeing next day delivery, addressed as follows:

         (a)      If to SCI, as Warrant Holder, at:

                           Address:       910 Louisiana, Suite 5000
                                          Houston, Texas 77002-4916
                           Attention:     Robert L. Roberts, Vice-President
                           Telecopier No. (713) 241-5222

         (b)      If to the Issuer at:

                           Brigham Exploration Company
                           6300 Bridge Point Parkway
                           Building 2, Suite 500
                           Austin, Texas 78730
                           Attention: President
                           Telecopier No.: (512) 427-3300

                  or at such other address as may be substituted by notice given
                  as herein provided. The giving of any notice required
                  hereunder may be waived in writing by the

                                       26
<PAGE>   30

                  party entitled to receive such notice. Every notice, demand,
                  request, consent, approval, declaration, delivery or other
                  communication hereunder shall be deemed to have been duly
                  given or served on the date on which personally delivered,
                  with receipt acknowledged, or three (3) Business Days after
                  the same shall have been deposited in the United States mail.

15.3     Indemnification. Except to the extent otherwise provided in Section 9.6
         of this Agreement, the Issuer agrees to indemnify and hold harmless
         Warrant Holder and its officers, directors, employees, agents,
         attorneys and Affiliates (each an "Indemnified Party") from and against
         any liabilities, obligations, losses, damages, penalties, actions,
         judgments, suits, claims, costs, attorneys' fees, expenses and
         disbursements of any kind which may be imposed upon, incurred by or
         asserted against such Indemnified Party relating to or arising out of
         (i) such Warrant Holder's exercise of the Warrants and/or ownership of
         any shares of Warrant Stock issued in consequence thereof, or (ii) any
         litigation to which such Warrant Holder is made a party in its capacity
         as a stockholder or Warrant Holder of the Issuer; provided, however,
         that the Issuer will not be liable hereunder to the extent that any
         liabilities, obligations, losses, damages, penalties, actions,
         judgments, suits, claims, costs, attorneys' fees, expenses or
         disbursements (A) arise solely from any violation by such Warrant
         Holder of any law or regulation applicable to it or (B) are found in a
         final non-appealable judgment by a court to have resulted from such
         Warrant Holder's bad faith or willful misconduct or violation of law.
         The procedures to be followed for claims of indemnification under this
         Section 15.3 shall be as set forth in Section 9.6(d) of this Agreement.

15.4     Remedies. Each Warrant Holder of Warrants and Warrant Stock, in
         addition to being entitled to exercise all rights granted by law,
         including recovery of damages, will be entitled to specific performance
         of its rights under Section 9 of this Agreement. The Issuer agrees that
         monetary damages would not be adequate compensation for any loss
         incurred by reason of a breach by it of the provisions of Section 9 of
         this Agreement, and hereby agrees to waive any defense to the contrary
         in any action for specific performance that a remedy at law would be
         adequate.

15.5     Successors and Assigns. Subject to the provisions of Sections 3.1 and 9
         of this Agreement, this Agreement and the rights evidenced hereby shall
         inure to the benefit of and be binding upon the successor of the Issuer
         and the successors and assigns of any Warrant Holder. The provisions of
         this Agreement are intended to be for the benefit of all Warrant
         Holders from time to time of the Warrants and Warrant Stock, and shall
         be enforceable by any such Warrant Holder.

15.6     Complete Agreement; Amendment. This Agreement, the Warrant
         Certificates, the Equity Conversion Agreement, the Credit Agreement and
         the Loan Documents constitute the complete agreement among the parties
         with respect to the subject matter hereof. This Agreement may be
         modified or amended or the provisions hereof waived only with the
         written consent of the Issuer and the Required Holders, provided that
         no Warrant may be modified or amended to reduce the number of shares of
         Common Stock for which such Warrant is exercisable or to increase the
         price at which such shares may be purchased upon exercise of such
         Warrant (before giving effect to any adjustment as provided

                                       27
<PAGE>   31

         herein) or to accelerate the Expiration Date without the prior written
         consent of the Warrant Holder thereof, and any amendment of Section 9
         of this Agreement shall also require the written consent of Warrant
         Holders of Warrants and/or Warrant Stock representing more than 50% of
         the total of (i) all shares of Warrant Stock then subject to purchase
         upon exercise of all Warrants then Outstanding, and (ii) all shares of
         Warrant Stock then Outstanding.

15.7     Severability. Wherever possible, each provision of this Agreement shall
         be interpreted in such manner as to be effective and valid under
         applicable law, but if any provision of this Agreement shall be
         prohibited by or invalid under applicable law, such provision shall be
         ineffective to the extent of such prohibition or invalidity, without
         invalidating the remainder of such provision or the remaining
         provisions of this Agreement.

15.8     Headings. The headings used in this Agreement are for the convenience
         of reference only and shall not, for any purpose, be deemed a part of
         this Agreement.

15.9     Governing Law; Consent to Jurisdiction and Venue. IN ALL RESPECTS,
         INCLUDING ALL MATTERS OF CONSTRUCTION, VALIDITY AND PERFORMANCE, THIS
         AGREEMENT AND THE OBLIGATIONS ARISING HEREUNDER SHALL BE GOVERNED BY,
         AND CONSTRUED AND ENFORCED IN ACCORDANCE WITH, THE LAWS OF THE STATE OF
         TEXAS APPLICABLE TO CONTRACTS MADE AND PERFORMED IN SUCH STATE, WITHOUT
         REGARD TO THE PRINCIPLES THEREOF REGARDING CONFLICT OF LAWS, AND ANY
         APPLICABLE LAWS OF THE UNITED STATES OF AMERICA.

15.10    Consent to Jurisdiction and Venue.

         (a)      THE ISSUER AND EACH WARRANT HOLDER HEREBY EXPRESSLY SUBMITS TO
                  THE JURISDICTION OF THE COURTS OF THE STATE OF TEXAS OR THE
                  UNITED STATES DISTRICT COURT FOR THE SOUTHERN DISTRICT OF
                  TEXAS. FINAL JUDGMENT AGAINST SUCH PARTY IN ANY SUCH SUIT
                  SHALL BE CONCLUSIVE, AND MAY BE ENFORCED IN ANY OTHER
                  JURISDICTIONS BY SUIT ON THE JUDGMENT OR AS OTHERWISE
                  PERMITTED BY APPLICABLE LAW, A CERTIFIED OR TRUE COPY OF WHICH
                  SHALL BE CONCLUSIVE EVIDENCE OF THE FACTS AND OF THE AMOUNT OF
                  ANY INDEBTEDNESS OR LIABILITY OF SUCH PARTY THEREIN DESCRIBED;
                  PROVIDED, HOWEVER, EACH PARTY MAY AT ITS OPTION BRING SUIT, OR
                  INSTITUTE OTHER JUDICIAL PROCEEDINGS AGAINST THE OTHER PARTY
                  OR ANY OF ITS ASSETS, IN THE COURTS OF ANY COUNTRY OR PLACE
                  WHERE SUCH PARTY OR SUCH ASSETS MAY BE FOUND.

         (b)      THE ISSUER AND EACH WARRANT HOLDER HEREBY IRREVOCABLY WAIVES
                  ANY OBJECTION WHICH IT MAY NOW OR HEREAFTER HAVE TO THE LAYING
                  OF VENUE OF ANY SUIT, ACTION OR PROCEEDING BROUGHT IN ANY
                  COURTS OF THE STATE OF TEXAS OR THE UNITED

                                       28
<PAGE>   32

                  STATES DISTRICT COURT FOR THE SOUTHERN DISTRICT OF TEXAS AND
                  HEREBY FURTHER IRREVOCABLY WAIVES ANY CLAIM THAT ANY SUCH
                  SUIT, ACTION OR PROCEEDING BROUGHT IN ANY SUCH COURT HAS BEEN
                  BROUGHT IN AN INCONVENIENT FORUM.

15.11    Counterparts: This Agreement may be executed in two or more
         counterparts, each of which shall be deemed an original, but all of
         which together shall constitute one and the same instrument.

                                       29
<PAGE>   33

         IN WITNESS WHEREOF, the parties hereto have caused this Agreement to be
duly executed as of the date first above written.

                                        BRIGHAM EXPLORATION COMPANY, as Issuer

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

                                        SHELL CAPITAL INC.,
                                        as Warrant Holder

                                        By:
                                           ------------------------------------
                                        Name:  Robert L. Roberts
                                        Title: Vice President

                                       30
<PAGE>   34

                                                                    EXHIBIT A TO
                                                               Warrant Agreement

                          (FORM OF WARRANT CERTIFICATE)

THE SECURITIES REPRESENTED BY THIS CERTIFICATE HAVE NOT BEEN REGISTERED UNDER
THE SECURITIES ACT OF 1933, AS AMENDED, OR ANY APPLICABLE STATE SECURITIES LAWS,
AND MAY NOT BE SOLD OR TRANSFERRED IN THE ABSENCE OF SUCH REGISTRATION OR AN
EXEMPTION THEREFROM. SUCH SECURITIES ARE SUBJECT TO THE RESTRICTIONS AND
PRIVILEGES SPECIFIED IN THE WARRANT AGREEMENT, DATED AS OF _____________, 2000,
BETWEEN BRIGHAM EXPLORATION COMPANY AND THE INITIAL HOLDER OF SECURITIES NAMED
THEREIN, A COPY OF WHICH IS ON FILE WITH THE SECRETARY OF BRIGHAM EXPLORATION
COMPANY AND WILL BE FURNISHED WITHOUT CHARGE TO THE HOLDER HEREOF UPON WRITTEN
REQUEST, AND THE HOLDER OF THIS CERTIFICATE AGREES TO BE BOUND THEREBY.

No. ___

                               WARRANT CERTIFICATE

This Warrant Certificate certifies that Shell Capital Inc., or registered
assigns thereof, is the holder of ________ warrants (the "Warrants") to purchase
shares of common stock of Brigham Exploration Company, a Delaware corporation
(the "Company"). Each Warrant entitles the holder, but only subject to the
conditions set forth herein and in the Warrant Agreement referred to below, to
purchase from the Company before 11:59 p.m., Central Standard Time, on the
Expiration Date, as such term is defined in the Warrant Agreement, one fully
paid and non-assessable share of common stock of the Company ("Warrant Stock")
at a price (the "Exercise Price") of _______________________ ($______) per share
of Warrant Stock payable as set forth in the Warrant Agreement. The number of
shares of Warrant Stock for which each Warrant is exercisable and the Exercise
Price are each subject to adjustment prior to the Expiration Date upon the
occurrence of certain events as set forth in the Warrant Agreement.

The Company may deem and treat the registered holders of the Warrants evidenced
hereby as the absolute owner thereof (notwithstanding any notation of ownership
or other writing hereon made by anyone), for the purpose of any exercise hereof
and of any distribution to the holders hereof, and for all other purposes.

Warrant Certificates, when surrendered at the office of the Company by the
registered holder hereof in person or by a legal representative duly authorized
in writing, may be exchanged, in the

                                     EXH A-1
<PAGE>   35

manner and subject to the limitations provided in the Warrant Agreement, but
without payment of any service charge, for another Warrant Certificate or
Warrant Certificates of like tenor evidencing in the aggregate a like number of
Warrants.

Upon due presentment for registration of transfer of this Warrant Certificate at
the office of the Company at 6300 Bridge Point Parkway, Building 2, Suite 500,
Austin, Texas 78730, Attention: President, or such other address as the Company
may specify in writing to the registered holder of the Warrants evidenced
hereby, a new Warrant Certificate or Warrant Certificates of like tenor and
evidencing in the aggregate a like number of Warrants shall be issued to the
transferee in exchange for this Warrant Certificate to the transferee(s) and, if
less than all the Warrants evidenced hereby are to be transferred, the
registered holder hereof, subject to the limitations provided in the Warrant
Agreement, without charge except for any tax or other governmental charge
imposed in connection therewith.

The Warrant Certificate is one of the Warrant Certificates referred to in the
Warrant Agreement, dated as of ____________, 2000, between the Company and the
initial holder of Warrants party thereto (the "Warrant Agreement"). Said Warrant
Agreement is hereby incorporated by reference in and made a part of this
instrument and is hereby referred to for a description of the rights, limitation
of rights, obligations, duties and immunities thereunder of the Company and the
holders, and in the event of any conflict between the terms of this Warrant
Certificate and the provisions of the Warrant Agreement, the provisions of the
Warrant Agreement shall control.

IN WITNESS WHEREOF, the Company has caused this Warrant Certificate to be duly
executed and its corporate seal to be impressed hereon and attached by its
secretary.

Dated:                        , 2000.
       -----------------------

                                        BRIGHAM EXPLORATION COMPANY

                                        By:
                                           ------------------------------
                                        Printed Name:
                                                     --------------------
                                        Title:
                                              ---------------------------

(CORPORATE SEAL)

ATTEST:

--------------------------------------
Secretary

                                    EXH A-2

<PAGE>   36

                                                                    Exhibit A To
                                                             Warrant Certificate

                                SUBSCRIPTION FORM

                 [To be executed only upon exercise of Warrants]

The undersigned registered owner of this Warrant Certificate irrevocably
exercises Warrants for the purchase of shares of Common Stock of [         ] and
herewith makes payment therefor

         $          in cash
          ----

all at the price and on the terms and conditions specified in the Warrant
Certificate and the Warrant Agreement, and requests that certificates for the
shares of Common Stock hereby purchased (and any securities or other property
issuable upon such exercise) be issued in the name of ____________________ and
delivered to _____________________________ whose address is
________________________________ and, if such shares of Common Stock shall not
include all of the shares of Common Stock issuable as provided in the Warrant
Certificate, that a new Warrant Certificate of like tenor and date for the
balance of the shares of Common Stock issuable thereunder be delivered to the
undersigned.

----------------------------------------------
(Name of Registered Owner)

----------------------------------------------
(Signature of Registered Owner)

----------------------------------------------
(Street Address)

----------------------------------------------
(City)         (State)         (Zip Code)

                                    EXH A-3

<PAGE>   37

                                                                    Exhibit B To
                                                             Warrant Certificate

                                 ASSIGNMENT FORM

FOR VALUE RECEIVED the undersigned registered owner of the attached Warrant
Certificate hereby sells, assigns and transfers unto the assignee named below
all of the rights of the undersigned under this Warrant Certificate, with
respect to the number of shares of Common Stock set forth below:

Name and Address of Assignee:
                                   -------------------------------

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

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

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

No. of Shares of
Common Stock
             ------

and does hereby irrevocably constitute and appoint_____________________________
attorney-in-fact to register such transfer on the books of [                   ]
maintained for that purpose, with full power of substitution in the premises.

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

Name:
        -------------------------------

Signature:
           --------------------------

Witness:
           --------------------------

The assignee named above hereby agrees to purchase and take the attached Warrant
Certificate pursuant to and in accordance with the terms and conditions of the
Warrant Agreement, dated as of ______________, 2000, between [                 ]
and the initial Holder named therein and agrees to be bound thereby.

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

Name:
      -------------------------------

Signature:
           --------------------------

                         Exh. B-1 to Warrant Certificate

<PAGE>   38

                                   SCHEDULE A

<TABLE>
<CAPTION>
Shares of
Common                                      Stock Options   Total
Stock                                       Outstanding     Warrants
Outstanding                      Vested     Unvested        Outstanding
As of February 16, 2000

<S>                              <C>        <C>             <C>
14,517,786                       360,671    1,485,390       2,000,000
</TABLE>

                                    SCH. A-1

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