THE MERIDIAN RESOURCE CORPORATION

================================================================================
CERTIFICATE OF DESIGNATION, PREFERENCES AND RIGHTS OF A SERIES OF PREFERRED
STOCK BY RESOLUTION OF THE BOARD OF DIRECTORS PROVIDING FOR AN ISSUE OF
3,982,906 SHARES OF PREFERRED STOCK DESIGNATED SERIES A CUMULATIVE CONVERTIBLE
PREFERRED STOCK.
================================================================================

      The Meridian Resource Corporation, a Texas Corporation (the "Company"),
pursuant to the provisions of Article 2.12 of the Texas Business Corporation Act
("TBCA"), does hereby state and certify that, pursuant to the authority
expressly vested in the Board of Directors of the Company by the Second Amended
and Restated Articles of Incorporation of the Company, as amended, that the
Board of Directors, at a meeting thereof duly called and held on March 27, 1998,
at which meeting a quorum was present and acting throughout, duly adopted the
following resolutions providing for the issue of shares of Preferred Stock
hereinafter referred to, and further providing with respect to such issue of
shares of Preferred Stock for such powers, designations, preferences and
relative, participating, optional and other special rights, and the
qualifications, limitations or restrictions thereof, as are hereinafter set
forth, in addition to those set forth in said Articles of Incorporation;

      RESOLVED, that pursuant to Article Four of the Second Amended and Restated
Articles of Incorporation of the Company, as amended (which authorizes the
Company to issue up to 25,000,000 shares of Preferred Stock, par value $1.00 per
share), the Board of Directors hereby provides for the issue of a series of up
to 3,982,906 shares of Preferred Stock designated "Series A Cumulative
Convertible Preferred Stock"; and

      RESOLVED, that the powers, designations, preferences and relative,
participating, optional and other special rights, and the qualifications,
limitations or restrictions thereof, of the shares of the Series A Cumulative
Convertible Preferred Stock shall be as follows:

      SECTION 1. DESIGNATION AND RANK. The designation of the series of
Preferred Stock created by this resolution shall be "Series A Cumulative
Convertible Preferred Stock", and the number of shares constituting this Series
shall have a stated value of $33.894850 per share (the "Stated Value"). The
shares of this Series shall rank prior to the Junior Stock (as defined in
Section 8) as to distribution of assets and payments of dividends.

      SECTION 2. DIVIDENDS.

      (a) Shares of this Series shall be entitled to receive, when and as
declared by the Board of Directors, a cash dividend at the dividend rate of four
percent (4%) per annum (the "Dividend Rate") on the Stated Value per share of
this Series, and no more; provided, however, that dividends shall cease to
accrue on shares of the Series on the following schedule: 1,327,635 shares on
the third anniversary of the date of original issuance (the "Sub-Series A-I");
1,327,636 shares on the fourth anniversary of the date of original issuance (the
"Sub-Series A-II"); and 1,327,635 shares on the fifth anniversary of the date of
original issuance (the "Sub-Series A-III") (collectively, the "Sub-Series"). The
certificates evidencing shares of the Series will specify whether the shares
represented thereby are designated Sub-Series A-I, Sub-Series A-I or Sub- Series
A-III. Dividends shall be cumulative, shall accrue (whether or not declared and
whether or not there shall be funds legally available for the payment of
dividends) from the data of original issuance and shall be payable in arrears,
out of assets legally available therefor, when and as declared by the Board of
Directors of the Company, on January 1, April 1, July 1, and October 1 of each
year in which dividends are payable, commencing October 1, 1998, (except that if
any such date is a Saturday, Sunday or a Business Day then such dividend shall
be payable without interest on the next day that is not a Saturday, Sunday or
Business Day) (each three-month period expiring on a dividend payment date being
referred to herein as a "Dividend Period"). Dividends shall be paid to the
holders of record of shares of each Sub-Series entitled thereto as they appear
on the stock register of the Company on such record dates, not exceeding 30 days
preceding the payment dates thereof, as shall be fixed by the Board of
Directors. Dividends on account of arrears for any past Dividend Periods (an
"Arrearage") may be declared and paid at any time, without reference to any
regular dividend payment date, to holders of record on such date, not exceeding
45 days preceding the payment date thereof, as may be fixed by the Board of
Directors. Until paid in full, each Arrearage shall also accrue dividends at the
rate of 4% per annum. All dividends and Arrearages respectively, shall be
declared and paid pro rata on all Sub-Series, based on the aggregate of the
accrued dividends, Arrearages and dividends on Arrearages, respectively, for the
various Sub-Series.

      (b) No dividends (other than a dividend in Junior Stock or other than as
provided in Section 2(b)) shall be declared or paid or set apart for payment on
Junior Stock for any period unless full cumulative dividends have been or
contemporaneously are declared and paid or declared and a sum sufficient for the
payment thereof set apart for such payment on this Series for all Dividend
Periods terminating on or prior to the date of payment of such full cumulative
dividends Holders of shares of this Series shall not be entitled to any
dividend, whether payable in cash, property or stock, in excess of full
cumulative dividends.

      (c) So long as any shares of this Series are outstanding, no dividend
(other than a dividend in Junior Stock or other than as provided in Section
2(b)) shall be declared or paid or set aside for payment or other distribution
declared or made upon the Junior Stock, nor shall any Junior Stock be redeemed,
purchased or otherwise acquired for any consideration (or any moneys be paid to
or made available for a 

                                      -2-

sinking fund for the redemption of any shares of Junior Stock) by the Company
(except by conversion into or in exchange for Junior Stock) unless, in each
case, full cumulative dividends on all outstanding shares of this Series then
payable shall have been paid.

      (d) Dividends payable on this Series for a period less than a full
Dividend Period shall be computed on the basis of the ratio of the number of
days in such partial period to the actual number of days in such full Dividend
Period.

      SECTION 3. VOTING AND ELECTION OF DIRECTORS.

      (a) The holders of outstanding shares of this Series shall be entitled to
vote, with the Common Stock and any other capital stock voting with the Common
Stock, with respect to all matters for which a vote of the holders of Common
Stock is taken. Each holder of outstanding shares of this Series shall be
entitled to that number of votes as are equal to the number of shares of this
Series held by such holder. The record date for holders of shares of the Series
entitled to vote on any matters submitted to the holders of Common Stock for
their vote, whether at an annual or special meeting of shareholders or by
unanimous written consent, shall be the same date as the record date established
for the Common Stock.

      (b) So long as any shares of this Series remain outstanding, the
affirmative vote or consent of the holders of a majority of the shares of this
Series outstanding at the time, voting as a single class, given in person or by
proxy, either in writing or at a meeting, shall be necessary to permit, effect
or validate any of the following: (i) the authorization, creation or issuance,
or any increase in the authorized or issued amount, of any class of series of
Senior Stock or Parity Stock (both as defined in Section 8) or (ii) the
amendment, restatement, modification, alteration or repeal of any of the
provisions of this Certificate of Designation.

      (c) Until the earlier of (i) the termination of the Stock Rights and
Restrictions Agreement to be entered into on or about June 30, 1998, as it may
be amended from time to time (the "SLOPI Agreement"), between the Company and
Shell Louisiana Onshore Properties, Inc., a Delaware Corporation ("SLOPI") or
(ii) SLOPI and its Affiliates shall Beneficially Own shares of Common Stock
constituting less than 21% of the then outstanding shares of Common Stock, then,
in connection with each election of directors of the Company, whether at an
annual or special meeting, the holders of this Series shall be entitled to elect
at such meeting a number of directors (the "Preferred Directors") such that,
after giving effect to the election of such persons to the Board of Directors of
the Company, the number of Preferred Directors then serving on the Board of
Directors of the Company shall equal the product (rounded downward to the
nearest whole number, but, in any event, not less than one) of (i) the total
number of directors constituting the entire Board of Directors multiplied by
(ii) 20% (the "Director Percentage").

      (d) If at any time the number of directors constituting the Board of
Directors of the Company shall decrease so that the holders of the Series would
be entitled to designate fewer directors than are then serving as Preferred
Directors, the holders of 

                                      -3-

the Series, one or more of the Preferred Directors shall resign so that the
percentage of the Board of Directors consisting of Preferred Directors does not
exceed the Director Percentage (rounded downward to the nearest whole number but
in no event less than one); provided, that, if a Preferred Director does not
resign, the members of the Board of Directors who are not Preferred Directors
shall be entitled to remove such Preferred Director. Further, (i) upon
termination of the SLOPI Agreement in accordance with its terms or (ii) the
conversion of all outstanding shares of this Series into Common Stock, all
Preferred Directors then serving as directors of the Company shall immediately
cease to be members of the Company's Board of Directors and shall be deemed to
have resigned on the date of such termination.

      (e) (i) In the event that any Preferred Director shall cease to serve as a
director for any reason (other than as set forth in Section (d) immediately
above), the vacancy resulting thereby shall be filled by appointment by any
Preferred Director remaining, and in the absence of action by the remaining
Preferred Directors, by the holders of the Series, and such Preferred Director
shall thereafter serve until the expiration of the term of the Preferred
Director replaced by such new Preferred Director.

          (ii) Subject to the provisions of Section 3(f) below, if there shall
exist at any time any vacancy or vacancies on the Board of Directors of the
Company as a result of any increase in the number of directors that constitutes
the entire Board of Directors of the Company, which the directors of the Company
then in office intend to fill in accordance with the Company's then existing
Articles of Incorporation, by-laws and applicable law, the holders of the Series
shall be entitled to designate one or more persons to fill such vacancy or
vacancies if and to the extent necessary so that, after giving effect to the
filling of such vacancy or vacancies, the number of Preferred Directors then
serving on the Board of Directors of the Company shall equal the Director
Percentage (rounded downward to the nearest whole number but in no event less
than one).

      (f) Notwithstanding anything to the contrary contained herein, no
Preferred Director may be a person who previously has been a director of the
Company and was properly removed for cause from the Board of Directors of the
Company or a person who has been convicted of a felony or a crime involving
moral turpitude.

      (g) At all times when there is a Preferred Director on the Company's Board
of Directors, at least one shall be a member of each Audit Committee of the
Board of Directors. All members of the Audit Committee shall have access to the
Company's independent accountants and all audit and tax work papers to the same
extent as any other member of the Audit Committee.

      (h) The Preferred Directors will be furnished with all information that is
provided to all other directors of the Company (in their capacities as such) at
the same time as such information is furnished to such other directors (in their
capacities as such).

                                      -4-

      (i) All Preferred Directors shall comply with the retirement policies of
the Company as in effect on the date hereof or as hereafter amended or modified
from time to time by the Board of Directors of the Company or its stockholders;
provided that no such amendment or modification to such policies shall be
binding upon a Preferred Director unless at least one Preferred Director shall
have voted in favor of such amendment or modification at the meeting, or in the
action in lieu of a meeting, of the Board of Directors of the Company at or in
which it is considered.

      (j) The election of Preferred Directors or approval under paragraph (b)
above shall be by holders of a majority of the issued and outstanding Series as
reflected on the stock books of the Company on the applicable record date, or if
by written consent, on the date of such consent. The record date for holders of
shares of the Series entitled to vote on matters pursuant to this Section 3 at a
meeting of the holders of the Series shall be established by the Board of
Directors of the Company and shall not precede 60 days before such meeting.
Special meetings of the holders of the Series may be called by any holder of
more than 25% of the outstanding shares of the Series, except that the Company
may call a special meeting of the holders of the Series in connection with any
action to be voted on pursuant to paragraph (b) above. Any action taken by
written consent of the holders of the Series need not be unanimous and such
action will be effective upon delivery of such written consent to the Secretary
of the Company.

      SECTION 4. LIQUIDATION. In the event of any complete liquidation,
dissolution or winding-up of the Company, whether voluntary or involuntary the
holders of shares of this Series shall each be entitled to receive out of assets
of the Company, whether such assets are capital or surplus, for each share of
this Series a sum equal to the Stated Value plus the amount of any accrued and
unpaid dividends on such share plus interest accrued but unpaid on any
Arrearage, before any distribution shall be made to the holders of Junior Stock
of the Company, and if the assets of the Company shall be insufficient to pay in
full such amounts, then such assets shall be distributed among the holders of
this Series ratably in accordance with the respective amounts that would be
payable on such shares if all amounts payable thereon were paid in full.

      SECTION 5. CONVERSION.

      (a) Each share of this Series shall be convertible at the option of the
record holder thereof at any time by presentation of the certificate
representing such share by the record holder in person or by registered mail,
return receipt requested with postage prepaid thereof, at the principal office
of the Company, and at such other offices, if any, as the Board of Directors may
determine, into the number of fully paid and nonassessable shares of Common
Stock determined by dividing the Stated Value by the Conversion Price in effect
on the Conversion Date.

      (b) The conversion price initially shall be $10.516125 (the "Conversion
Price") and shall be subject to adjustment from time to time as follows:

                                      -5-

            (i) If the Company, at any time while any shares of this Series are
      outstanding, shall (A) pay a stock dividend or stock dividends or
      otherwise make a distribution or distributions on shares of its capital
      stock payable in shares of Common Stock (or in securities convertible into
      shares of Common Stock), (B) except as set forth in clause (A) above, pay
      a stock dividend or make a distribution on shares of its capital stock
      payable in shares of its capital stock of any class other than Common
      Stock or a class convertible into Common Stock, (C) subdivide outstanding
      shares of Common Stock into a larger number of shares, (D) combine
      outstanding shares of Common Stock into a smaller number of shares, or (E)
      issue by reclassification of shares of Common Stock any shares of capital
      stock of the Company of any class or classes, the Conversion Price in
      effect immediately prior to such action shall be adjusted so that the
      holder of any shares of this Series thereafter surrendered for conversion
      shall be entitled to receive the number and class or classes of shares of
      the capital stock of the Company which he would have owned or have been
      entitled to receive immediately after the happening of any of the events
      described above, had such shares of this Series been converted on or
      immediately prior to the record date for such dividend or distribution or
      the effective date of such subdivision, combination or reclassification,
      as the case may be. An adjustment made pursuant to this subsection 5(b)(i)
      shall become effective immediately after the record date in the case of a
      dividend or distribution and shall become effective immediately after the
      effective date in the case of a subdivision, combination or
      reclassification.

            (ii) If the Company, at any time while any shares of this Series are
      outstanding, shall issue rights or warrants to all holders of Common Stock
      entitling them (for a period expiring within 45 days after the record date
      mentioned below) to subscribe for or purchase shares of Common Stock at a
      price per share less than the Per Share Market Value of Common Stock at
      the record date mentioned below, the Conversion Price at which each share
      of this Series shall thereafter be convertible shall be reduced by
      multiplying the Conversion Price in effect immediately prior to such
      record date by a fraction, of which the denominator shall be the number of
      shares of Common Stock (excluding treasury shares, if any) outstanding on
      the date of issuance of such rights or warrants plus the number of
      additional shares of Common Stock offered for subscription or purchase,
      and of which the numerator shall be the number of shares of Common Stock
      (excluding treasury shares, if any) outstanding on the date of issuance of
      such rights or warrants plus the number of shares which the aggregate
      offering price of the total number of shares so offered would purchase at
      such Per Share Market Value. Such adjustment shall be made whenever such
      rights or warrants are issued, and shall become effective immediately
      after the record date for the determination of stockholders entitled to
      receive such rights or warrants. However, upon the expiration of any right
      or warrant to purchase Common Stock the issuance of which resulted in an
      adjustment in the Conversion Price of the shares of this Series pursuant
      to this Subsection 5(b)(ii), if any such right or warrant shall expire and
      shall not have been fully exercised, the Conversion Price per share of
      Common Stock at which 

                                      -6-

      each share of this Series shall thereafter be convertible shall
      immediately upon such expiration be recomputed and effective immediately
      upon such expiration be increased to the price which it would have been
      (but reflecting any other adjustments in the Conversion Price made
      pursuant to the provisions of this Section 6 after the issuance of such
      rights or warrants) had the adjustment of the Conversion Price made upon
      the issuance of such rights or warrants been made on the basis of offering
      for subscription or purchase only that number of shares of Common Stock
      actually purchased upon the exercise of such rights or warrants which were
      actually exercised. No adjustment shall be made pursuant to this Section
      5(b)(ii) if rights and warrants are also distributed to the holders of
      this Series on the basis of the number of shares of Common Stock then
      issuable on conversion of the shares of this Series.

            (iii) If the Company, at any time while shares of this Series are
      outstanding, shall distribute to all holders of Common Stock evidences of
      its indebtedness or assets (excluding cash dividends or cash distributions
      paid out of earned surplus) or rights or warrants to subscribe for or
      purchase any security (excluding those referred to in Subsection 5(b)(ii)
      above) then in each such case the Conversion Price per share of Common
      Stock at which each share of this Series shall thereafter be convertible
      shall be determined by multiplying the Conversion Price in effect
      immediately prior to the record date fixed for determination of
      stockholders entitled to receive such distribution by a fraction, of which
      the denominator shall be the Per Share Market Value of Common Stock
      determined as of the record date mentioned above, and of which the
      numerator shall be such Per Share Market Value of the Common Stock, less
      the then fair market value (as determined by the Board of Directors of the
      Company (the "Board") in good faith, whose determination shall be
      conclusive if made in good faith; provided, however, that in the event of
      a distribution or series of related distributions exceeding 10% of the net
      assets of the Company, then such fair market value shall be determined by
      a nationally recognized or major regional investment banking firm or firm
      of independent certified public accountants of recognized standing (which
      may be the firm that regularly examines the financial statements of the
      Company) selected in good faith by the Board, and in either case shall be
      described in a statement provided to all registered holders of this
      Series) of the portion of assets or evidences of indebtedness so
      distributed or such subscription rights applicable to one share of Common
      Stock. Such adjustment shall be made whenever any such distribution is
      made and shall become effective immediately after the record date
      mentioned above. No adjustment shall be made pursuant to this subsection
      5(b)(iii) if such evidence of indebtedness or assets are also distributed
      to the holders of this Series on the basis of the number of shares of
      Common Stock then issuable on the conversion of the shares of this Series.

            (iv) No notification to the holders of any adjustment in the
      Conversion Price otherwise required by this Section 5 shall be required
      unless such adjustment would require an increase or decrease of at least
      1% in such price; provided, however, that any adjustment which by reason
      of this 

                                      -7-

      subsection 5(b)(iv) is not required to be made shall be carried forward
      and taken into account in any subsequent adjustments, and that upon
      presentment of shares of this Series for conversion, all adjustment shall
      be made calculating the conversion rights of such holder. All calculations
      under this Section 5 shall be made to the nearest cent or the nearest
      1/100th of a share, as the case may be.

            (v) Subject to (iv) above, whenever the Conversion Price is
      adjusted, as herein provided, the Company shall promptly mail to each
      registered holder of shares of this Series a notice setting forth the
      Conversion Price after such adjustment and setting forth a brief statement
      of the facts requiring such adjustment. Such notice prepared in good faith
      shall be conclusive evidence of the correctness of such adjustment absent
      manifest error.

            (vi) In case:

                  (A) the Company shall declare a dividend (or any other
            distribution) on the Common Stock payable otherwise than in cash out
            of its earned surplus; or

                  (B) the Company shall declare a special nonrecurring cash
            dividend on or a redemption of its Common Stock; or

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

                  (D) the approval of any stockholders of the Company shall be
            required in connection with any reclassification of the Common Stock
            of the Company (other than a subdivision or combination of the
            outstanding shares of Common Stock), any consolidation or merger to
            which the Company is a party, any sale or transfer of all or
            substantially all of the assets of the Company, or any compulsory
            share exchange whereby the Common Stock is converted into other
            securities, cash or property; or

                  (E) of the voluntary or involuntary dissolution, liquidation
            or winding up of the affairs of the Company,

then the Company shall cause to be filed at each office or agency maintained for
the purpose of conversion of the shares of this series, and shall cause to be
mailed to the holders of record of the shares of this Series at their last
addresses as they shall appear upon the stock books of the Company, at least 10
days prior to the applicable record date hereinafter specified, a notice stating
(x) the date on which a record is to be taken for the purpose of such dividend,
distribution, redemption, 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, redemption, rights or warrants are to be
determined, or (y) the date on which such reclassification, consolidation,
merger, sale, transfer, share exchange, dissolution, liquidation or winding up
is expected to become 

                                      -8-

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, transfer, share exchange, dissolution, liquidation
or winding up (but no failure to mail such notice or any defect therein or in
the mailing thereof shall affect the validity of the corporate action required
to be specified in such notice).

      (c) In case of any reclassification of the Common Stock, then the holders
of the shares of this Series then outstanding shall have the right thereafter to
convert such shares only into the kind and amount of shares of stock and other
securities and property receivable upon or deemed to be held following such
reclassification by a holder of a number of shares of the Common Stock of the
Company into which such shares of this Series could have been converted
immediately prior to such reclassification. This provision shall similarly apply
to successive reclassifications.

      (d) In case of any consolidation or merger of the Company with or into
another Person in which the Company is not the surviving entity or any
compulsory share exchange pursuant to any of which the Common Stock is converted
into other securities, cash or property (any such event being hereinafter
referred to as "Reorganization"), then the terms of such Reorganization shall
provide that each holder of share of this Series then outstanding shall have the
right to receive in exchange therefor the kind and amount of shares of stock and
other securities and property receivable upon such Reorganization
("Reorganization Consideration") by a holder of the number of shares of the
Common Stock of the Company into which (x) a share of this Series (or
Sub-Series) could have been converted as of the effective date of the
Reorganization, plus (y) the Arrearage (if any) on a share of this Series could
have been exchanged as of the effective date of the Reorganization.

      (e) In case at any time conditions shall arise by reason of action taken
by the Company, which, in the opinion of the Board of Directors of the Company,
are not adequately covered by the other provisions hereof and which might
materially and adversely affect the rights of the holders of shares of this
Series, the Board of Directors of the Company shall appoint a firm of
independent certified public accountants of recognized standing (which may be
the firm that regularly examines the financial statements of the Company), who
shall give their opinion as to the adjustment, if any (not inconsistent with the
standards established in this Section 5, of the Conversion Price (including, if
necessary, any adjustment as to the securities into which shares of this Series
may thereafter be convertible) which is or would be required to preserve the
rights of the holders of shares of this Series. The Board of Directors of the
Company shall make the adjustment recommended forthwith upon the receipt of such
opinion or the taking of any such action contemplated, as the case may be;
provided, however, that no such adjustment of the Conversion Price shall be made
which in the opinion of the investment banking firm or firm of accountants
giving the aforesaid opinion would result in an increase of the Conversion Price
to more than the Conversion Price then in effect.

                                      -9-

      Section 6. MATTERS RELATING TO ISSUANCE OF COMMON STOCK. The following
provisions shall be applicable to issuances of Common Stock upon conversion of
shares of this Series.

      (a) The Company covenants that it will at all times reserve and keep
available, out of its authorized and unissued Common Stock solely for the
purpose of issuance upon conversion of this Series as herein provided, free from
preemptive rights or any other actual or contingent purchase rights of Persons
other than the holders of shares of this Series, such number of shares of Common
Stock as shall then be issuable upon the conversion of all outstanding shares of
this Series. The Company covenants that all shares of Common Stock that shall be
so issuable shall upon issue be duly and validly issued and fully paid and
nonassessable.

      (b) The Company shall not be required to issue stock certificates
representing fractions of shares of Common Stock, but may, if otherwise
permitted, make a cash payment in respect of any final fraction of a share based
on the Per Share Market Value at such time. If the Company elects not, or is
unable, to make such a cash payment, the holder of a share of this Series shall
be entitled to receive, in lieu of the final fraction of a share, one whole
share of Common Stock.

      (c) The issuance of certificates for shares of Common Stock on conversion
of this Series shall be made without charge to the holders thereof for any
documentary stamp or similar taxes that may be payable in respect of the issue
or delivery of such certificate, provided, that the Company shall not be
required to pay any tax that may be payable in respect of any transfer involved
in the issuance and delivery of any such certificate in a name other than that
of the holder of the shares of this Series converted was made and the Company
shall not be required to issue or deliver such certificates unless or until the
Person or Persons requesting the issuance thereof shall have paid to the Company
the amount of such tax or shall have established to the satisfaction of the
Company that such tax has been paid.

      (d) The exercise by a holder of shares of this Series of the conversion
rights granted herein is subject in all respects to and conditioned upon
compliance by the parties with the HSR Act, and rules and regulations
promulgated pursuant thereto, to the extent that said act, rules and regulations
are applicable to such exercise. The Company and such holder agree to make such
filings with and provide such information to the Federal Trade Commission and
the Department of Justice with respect to such exercise as are required in
connection with the HSR Act in a timely manner and to join each others request
for early termination. The Company and such holder will use such reasonable
efforts to obtain all governmental approval required to permit such exercise and
to cause early termination of the waiting period under the HSR Act.

      SECTION 7. AUTOMATIC CONVERSION. If on or after June 30, 2001 (a) any
shares of this Series have not been converted into shares of Common Stock and
(b) the mean average Per Share Market Value exceeds 150% of the Conversion Price
for 75 consecutive Trading Days, then all such shares of this Series shall
automatically be converted into the number of shares of Common Stock determined
by dividing the 

                                      -10-

Stated Value by the Conversion Price in effect at the time of conversion;
provided that SLOPI and its Affiliates may elect to retain an aggregate of one
share of Preferred Stock that will not be automatically converted pursuant to
this Section 7.

      SECTION 8. DEFINITIONS. For the purposes hereof, the following terms shall
have the following respective meanings:

      "Affiliate" shall mean, with respect to any specified Person, any other
Person, directly or indirectly controlling or controlled by or under direct or
indirect common control with such specified Person. For the purposes of this
definition, "control" (including, with correlative meanings, "controlling,"
"controlled by," and "under common control with") means the power to direct or
cause the direction of the management and policies of such Person, directly or
indirectly, whether through the ownership of voting securities, by contract, or
otherwise and, and with respect to a corporation or partnership, control shall
mean direct or indirect ownership of more than fifty percent (50%) of the voting
stock or general partnership interest or voting interest in any such corporation
or partnership.

      "Arrearage" has the meaning specified in Section 2(a).

      "Beneficially Own" shall have the meaning assigned to such term in Rule
13d-3 under the Exchange Act in effect on the date hereof. "Beneficial Owner"
and "Beneficial Ownership" shall have correlative meanings.

      "Business Day" shall mean any day that commercial banks located in
Houston, Texas are legally open for business.

      "Change of Control" shall mean the acquisition by a Person other than
SLOPI or its Affiliates of Beneficial Ownership of more than 50% of the then
outstanding shares of Common Stock.

      "Common Stock" means shares now or hereafter authorized of the class of
Common Stock, $0.01 par value, of the Company presently authorized and stock of
any other class into which such shares may hereafter have been reclassified or
changed.

      "Conversion Date" means the date the stock certificate is received by the
Company for conversion in accordance with Section 5(a).

      "Conversion Price" has the meaning specified in Section 5(b).

      "Dividend Period" has the meaning specified in Section 2(a).

      "Dividend Rate" has the meaning specified in Section 2(a).

      "Fully Diluted Shares" means, at any time, the sum of (i) the shares of
Common Stock then outstanding plus (ii) the number of shares of Common Stock
reserved for issuance or issuable in connection with the exercise, exchange or
conversion of options, 

                                      -11-

warrants or securities of the Company then outstanding which are exercisable or
exchangeable for shares of Common Stock or are convertible into shares of Common
Stock (including, without limitation, this Series).

      "HSR Act" means the Hart-Scott-Rodino Antitrust Improvements Act of
1976, as amended.

      "Junior Stock" means the Common Stock of the Company and any other stock
of the Company over which shares of this Series has a preference as to
distribution of assets and payment of dividends.

      "Parity Stock" means any stock of the Company ranking as to distribution
of assets and payment of dividends on a parity with this Series.

      "Per Share Market Value" means on any particular date (a) the last sale
price per share of the Common Stock on such date on the principal stock exchange
on which the Common Stock has been listed or, if there is no such price on such
date, then the last sale price on such exchange on the date nearest preceding
such date, or (b) if the Common Stock is not listed on any stock exchange, the
final bid price for a share of Common Stock in the over-the-counter market, as
reported by The Nasdaq Stock Market at the close of business on such date, or
the last sales price if such price is reported and final ibid prices are not
available, or (c) if the Common Stock is not quoted on The Nasdaq Stock Market,
the bid price for a share of Common Stock in the over-the-counter market as
reported by the National Quotation Bureau Incorporated (or any similar
organization or agency succeeding to its functions of reporting prices), or (d)
if the Common Stock is no longer publicly traded, as determined by one of the
investment banking firms listed on Schedule I to the SLOPI Agreement selected in
good faith by the Board of Directors of the Company, provided, that none of the
transactions related to the foregoing shall include purchases by any "affiliate"
(as such term is defined in the General Rules and Regulations under the
Securities Act of 1933) of the Company.

      "Person" means any individual, firm, partnership, association, group (as
such term is defined in Section 13(d)(3) of the Exchange Act, as in effect on
the date hereof), corporation, trust, business trust or other entity, and
includes any successor (by merger or otherwise) of any such entity.

      "Preferred Stock" means the Company's Preferred Stock, par value $0.01
per share.

      "Reorganization" has the meaning specified in Section 5(d).

      "Reorganization Consideration" has the meaning specified in
Section 5(d).

      "Senior Stock" means any stock of the Company which has a priority over
shares of this Series as to payment of dividends or distribution of assets of
the Company.

                                      -12-

      "SLOPI" means Shell Louisiana Onshore Properties, Inc., a Delaware
corporation.

      "SLOPI Agreement" has the meaning assinged thereto in Section 3.C.

      "Stated Value" has the meaning specified in Section 1.

      "Trading Day" means (a) a day on which the Common Stock is traded on the
principal stock exchange on which the Common Stock has been listed, or (b) if
the Common Stock is not listed on any stock exchange, a day on which the Common
Stock is quoted in the over-the-counter market, as reported by National
Association of Securities Dealers Automated Quotation System ("NASDAQ"), or (c)
if the Common Stock is not quoted by NASDAQ, a day on which the Common Stock is
quoted in the over-the-counter market as reported by the National Quotation
Bureau Incorporated (or any similar organization or agency succeeding to its
functions of reporting prices).


      IN WITNESS WHEREOF, The Meridian Resource Corporation has caused this
Certificate to be signed by a duly authorized officer, this 30th day of June,
1998.

                              THE MERIDIAN RESOURCE CORPORATION

                              By: /s/ JOSEPH A. REEVES, JR.
                                      Joseph A. Reeves, Jr.
                                      Chief Executive Officer and
                                      Chairman of the Board
ATTEST:

By:  M. MELINDA FREW

                                      -13-