Exhibit 24.1

                               POWER OF ATTORNEY

                                 April 12, 2018

        Know all by these presents, that the undersigned hereby constitutes and
appoints Jack Hightower, Steven W. Tholen and Rodney L. Woodard of Pure
Acquisition Corp. (the "Company") or any of them signing singly, and with full
power of substitution, the undersigned's true and lawful attorney-in-fact to:

    1.  prepare, execute in the undersigned's name and on the undersigned's
        behalf, and submit to the U.S. Securities and Exchange Commission (the
        "SEC") a Form ID, including amendments thereto, and any other documents
        necessary or appropriate to obtain codes and passwords enabling the
        undersigned to make electronic filings with the SEC of reports required
        by Section 16(a) of the Securities Exchange Act of 1934 or any rule or
        regulation of the SEC;

    2.  execute for and on behalf of the undersigned with respect to the
        Company, Schedules 13D and 13G and Forms 3, 4, and 5 in accordance with
        Sections 13 and 16(a) of the Securities Exchange Act of 1934 and the
        rules thereunder;

    3.  do and perform any and all acts for and on behalf of the undersigned
        which may be necessary or desirable to complete and execute any such
        Schedules 13D or 13G or Form 3, 4, or 5, complete and execute any
        amendment or amendments thereto, and timely file such form with the SEC
        and any stock exchange or similar authority; and

    4.  take any other action of any type whatsoever in connection with the
        foregoing which, in the opinion of such attorney-in-fact, may be of
        benefit to, in the best interest of, or legally required by, the
        undersigned, it being understood that the documents executed by such
        attorney-in-fact on behalf of the undersigned pursuant to this Power of
        Attorney shall be in such form and shall contain such terms and
        conditions as such attorney-in-fact may approve in such attorney-in-
        fact's discretion.

        The undersigned hereby grants to each such attorney-in-fact full power
and authority to do and perform any and every act and thing whatsoever
requisite, necessary, or proper to be done in the exercise of any of the rights
and powers herein granted, as fully to all intents and purposes as the
undersigned might or could do if personally present, with full power of
substitution or revocation, hereby ratifying and confirming all that such
attorney-in-fact, or such attorney-in-fact's substitute or substitutes, shall
lawfully do or cause to be done by virtue of this power of attorney and the
rights and powers herein granted. The undersigned acknowledges that the
foregoing attorneys-in-fact, in serving in such capacity at the request of the
undersigned, are not assuming, nor is the Company assuming any of the
undersigned's responsibilities to comply with Sections 13 and 16 of the
Securities Exchange Act of 1934.

        This Power of Attorney shall remain in full force and effect until the
undersigned is no longer required to file Forms 3, 4, and 5 with respect to the
undersigned's holdings of and transactions in securities issued by the Company,
unless earlier revoked by the undersigned in a signed writing delivered to the
foregoing attorneys-in-fact.

                                [Signature page follows]

        IN WITNESS WHEREOF, the undersigned has caused this Power of Attorney to
be executed as of the date first written above.

                                        SYLVIA K. BARNES

                                        /s/ Sylvia K. Barnes
                                        ---------------------------------------
                                        Name: Sylvia K. Barnes


                                        GREG COLVIN

                                        /s/ Greg Colvin
                                        ---------------------------------------
                                        Name: Greg Colvin


                                        JARED S. STURDIVANT

                                        /s/ Jared S. Sturdivant
                                        ---------------------------------------
                                        Name: Jared S. Sturdivant


                                        STEVEN W. THOLEN

                                        /s/ Steven W. Tholen
                                        ---------------------------------------
                                        Name: Steven W. Tholen


                                        RODNEY L. WOODARD

                                        /s/ Rodney L. Woodard
                                        ---------------------------------------
                                        Name: Rodney L. Woodard


                                        JACK HIGHTOWER

                                        /s/ Jack Hightower
                                        ---------------------------------------
                                        Name: Jack Hightower