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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