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 <front>
  <journal-meta>
   <journal-id journal-id-type="publisher-id">Russian Journal of Earth Sciences</journal-id>
   <journal-title-group>
    <journal-title xml:lang="en">Russian Journal of Earth Sciences</journal-title>
    <trans-title-group xml:lang="ru">
     <trans-title>Russian Journal of Earth Sciences</trans-title>
    </trans-title-group>
   </journal-title-group>
   <issn publication-format="online">1681-1208</issn>
  </journal-meta>
  <article-meta>
   <article-id pub-id-type="publisher-id">57771</article-id>
   <article-id pub-id-type="doi">10.2205/2023ES000857</article-id>
   <article-id pub-id-type="edn">bycury</article-id>
   <article-categories>
    <subj-group subj-group-type="toc-heading" xml:lang="ru">
     <subject>ОРИГИНАЛЬНЫЕ СТАТЬИ</subject>
    </subj-group>
    <subj-group subj-group-type="toc-heading" xml:lang="en">
     <subject>ORIGINAL ARTICLES</subject>
    </subj-group>
    <subj-group>
     <subject>ОРИГИНАЛЬНЫЕ СТАТЬИ</subject>
    </subj-group>
   </article-categories>
   <title-group>
    <article-title xml:lang="en">Mathematical Model of Freezing of Rocks Saturated With Salt Solution Taking Into Account the Influence of Osmosis</article-title>
    <trans-title-group xml:lang="ru">
     <trans-title>Mathematical Model of Freezing of Rocks Saturated With Salt Solution Taking Into Account the Influence of Osmosis</trans-title>
    </trans-title-group>
   </title-group>
   <contrib-group content-type="authors">
    <contrib contrib-type="author">
     <contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-2468-1272</contrib-id>
     <name-alternatives>
      <name xml:lang="ru">
       <surname>Рамазанов</surname>
       <given-names>Мукамай Магомедович</given-names>
      </name>
      <name xml:lang="en">
       <surname>Ramazanov</surname>
       <given-names>Mukamay Magomedovich</given-names>
      </name>
     </name-alternatives>
     <email>mukamay-ipg@mail.ru</email>
     <bio xml:lang="ru">
      <p>доктор физико-математических наук;</p>
     </bio>
     <bio xml:lang="en">
      <p>doctor of physical and mathematical sciences;</p>
     </bio>
     <xref ref-type="aff" rid="aff-1"/>
     <xref ref-type="aff" rid="aff-2"/>
    </contrib>
    <contrib contrib-type="author">
     <contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-4224-1246</contrib-id>
     <name-alternatives>
      <name xml:lang="ru">
       <surname>Булгакова</surname>
       <given-names>Наталья Сергеевна</given-names>
      </name>
      <name xml:lang="en">
       <surname>Bulgakova</surname>
       <given-names>Natal'ya Sergeevna</given-names>
      </name>
     </name-alternatives>
     <email>ipgnatali@mail.ru</email>
     <bio xml:lang="ru">
      <p>кандидат технических наук;</p>
     </bio>
     <bio xml:lang="en">
      <p>candidate of technical sciences;</p>
     </bio>
     <xref ref-type="aff" rid="aff-3"/>
     <xref ref-type="aff" rid="aff-4"/>
    </contrib>
    <contrib contrib-type="author">
     <contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-8033-8452</contrib-id>
     <name-alternatives>
      <name xml:lang="ru">
       <surname>Лобковский</surname>
       <given-names>Леопольд Исаевич</given-names>
      </name>
      <name xml:lang="en">
       <surname>Lobkovsky</surname>
       <given-names>Leopold Isaevich</given-names>
      </name>
     </name-alternatives>
     <email>llobkovsky@mail.ru</email>
     <xref ref-type="aff" rid="aff-5"/>
     <xref ref-type="aff" rid="aff-6"/>
    </contrib>
   </contrib-group>
   <aff-alternatives id="aff-1">
    <aff>
     <institution xml:lang="ru">Институт проблем геотермии и возобновляемой энергетики  - филиал Объединенного института высоких температур РАН</institution>
     <country>Россия</country>
    </aff>
    <aff>
     <institution xml:lang="en">Institute for Geothermal Research and Renewable Energy (Branch of the Joint Institute for High Temperatures, Russian Academy of Sciences)</institution>
     <country>Russian Federation</country>
    </aff>
   </aff-alternatives>
   <aff-alternatives id="aff-2">
    <aff>
     <institution xml:lang="ru">Институт динамики геосфер РАН</institution>
     <country>Россия</country>
    </aff>
    <aff>
     <institution xml:lang="en">Institute of Geosphere Dynamics of the Russian Academy of Sciences</institution>
     <country>Russian Federation</country>
    </aff>
   </aff-alternatives>
   <aff-alternatives id="aff-3">
    <aff>
     <institution xml:lang="ru">Институт проблем геотермии и возобновляемой энергетики  - филиал Объединенного института высоких температур РАН</institution>
     <city>Махачкала</city>
     <country>Россия</country>
    </aff>
    <aff>
     <institution xml:lang="en">Institute for Geothermal Research and Renewable Energy (Branch of the Joint Institute for High Temperatures, Russian Academy of Sciences)</institution>
     <city>Makhachkala</city>
     <country>Russian Federation</country>
    </aff>
   </aff-alternatives>
   <aff-alternatives id="aff-4">
    <aff>
     <institution xml:lang="ru">Дагестанский государственный институт народного хозяйства</institution>
     <country>Россия</country>
    </aff>
    <aff>
     <institution xml:lang="en">Dagestan State Institute of National Economy</institution>
     <country>Russian Federation</country>
    </aff>
   </aff-alternatives>
   <aff-alternatives id="aff-5">
    <aff>
     <institution xml:lang="ru">Институт океанологии им. П.П. Ширшова РАН</institution>
    </aff>
    <aff>
     <institution xml:lang="en">P.P.Shirshov Institute of Oceanology of the Russian Academy of Science</institution>
    </aff>
   </aff-alternatives>
   <aff-alternatives id="aff-6">
    <aff>
     <institution xml:lang="ru">Томский государственный университет</institution>
    </aff>
    <aff>
     <institution xml:lang="en">Tomsk State University</institution>
    </aff>
   </aff-alternatives>
   <pub-date publication-format="print" date-type="pub" iso-8601-date="2023-12-15T00:00:00+03:00">
    <day>15</day>
    <month>12</month>
    <year>2023</year>
   </pub-date>
   <pub-date publication-format="electronic" date-type="pub" iso-8601-date="2023-12-15T00:00:00+03:00">
    <day>15</day>
    <month>12</month>
    <year>2023</year>
   </pub-date>
   <volume>23</volume>
   <issue>5</issue>
   <fpage>1</fpage>
   <lpage>15</lpage>
   <history>
    <date date-type="received" iso-8601-date="2023-03-20T00:00:00+03:00">
     <day>20</day>
     <month>03</month>
     <year>2023</year>
    </date>
    <date date-type="accepted" iso-8601-date="2023-06-22T00:00:00+03:00">
     <day>22</day>
     <month>06</month>
     <year>2023</year>
    </date>
   </history>
   <self-uri xlink:href="https://rjes.ru/en/nauka/article/57771/view">https://rjes.ru/en/nauka/article/57771/view</self-uri>
   <abstract xml:lang="ru">
    <p>The paper presents a mathematical model of rocks freezing saturated with salt solution under impact of osmotic force. Osmosis is related to the salt concentration gradient, which is characteristic for solutions, and it is a powerful mechanism for the movement of solutions in poorly permeable porous media. A mathematical criterion for the formation of closed “pockets” with brines (cryopags) in frozen rocks has been obtained. This criterion is shown to be significantly depends on the osmosis coefficient. The model includes three layers of a porous medium saturated, respectively, with ice, ice and solution, and salt solution only. A special case was studied when there is only a second layer with a movable boundary, on which a phase transition from the second layer to the third one occurs. The investigated layer is saturated with a salt solution and ice in thermodynamic equilibrium. Other layers are replaced by boundary conditions. An approximate analytical solution of the problem is found in a self-similar formulation. The nature of the influence of osmotic force on the freezing process of rocks saturated with solution is shown. The characteristic patterns associated with the considered process are revealed. One of the features of the osmosis influence is the fact that it can cause the movement (migration) of the solution in the direction of increasing pressure, i.e. in the direction opposite to the driving force caused by the pressure gradient.</p>
   </abstract>
   <trans-abstract xml:lang="en">
    <p>The paper presents a mathematical model of rocks freezing saturated with salt solution under impact of osmotic force. Osmosis is related to the salt concentration gradient, which is characteristic for solutions, and it is a powerful mechanism for the movement of solutions in poorly permeable porous media. A mathematical criterion for the formation of closed “pockets” with brines (cryopags) in frozen rocks has been obtained. This criterion is shown to be significantly depends on the osmosis coefficient. The model includes three layers of a porous medium saturated, respectively, with ice, ice and solution, and salt solution only. A special case was studied when there is only a second layer with a movable boundary, on which a phase transition from the second layer to the third one occurs. The investigated layer is saturated with a salt solution and ice in thermodynamic equilibrium. Other layers are replaced by boundary conditions. An approximate analytical solution of the problem is found in a self-similar formulation. The nature of the influence of osmotic force on the freezing process of rocks saturated with solution is shown. The characteristic patterns associated with the considered process are revealed. One of the features of the osmosis influence is the fact that it can cause the movement (migration) of the solution in the direction of increasing pressure, i.e. in the direction opposite to the driving force caused by the pressure gradient.</p>
   </trans-abstract>
   <kwd-group xml:lang="ru">
    <kwd>osmosis</kwd>
    <kwd>freezing</kwd>
    <kwd>porous medium</kwd>
    <kwd>salt solution</kwd>
    <kwd>filtration</kwd>
    <kwd>phase transition</kwd>
   </kwd-group>
   <kwd-group xml:lang="en">
    <kwd>osmosis</kwd>
    <kwd>freezing</kwd>
    <kwd>porous medium</kwd>
    <kwd>salt solution</kwd>
    <kwd>filtration</kwd>
    <kwd>phase transition</kwd>
   </kwd-group>
   <funding-group>
    <funding-statement xml:lang="ru">This research, in general, was funded by the Russian Science Foundation, grant No. 22-67-00025.</funding-statement>
    <funding-statement xml:lang="en">This research, in general, was funded by the Russian Science Foundation, grant No. 22-67-00025.</funding-statement>
   </funding-group>
  </article-meta>
 </front>
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