{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T20:19:58Z","timestamp":1787170798002,"version":"build-2736575974"},"reference-count":53,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T00:00:00Z","timestamp":1772236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Gels"],"abstract":"<jats:p>The aim of this work is to describe the gelation process of a crosslinked polymer composition depending on its flow rate in free space and in pore space. The object of the study is a polymer solution based on partially hydrolyzed polyacrylonitrile and chromium acetate. A team of researchers has proposed a new approach to describing the kinetic viscosity curve of a crosslinked polymer system in a free volume. This approach takes into account oscillatory variations in the structural and mechanical characteristics relative to a smoothly increasing gelation curve. The nonlinear effects are linked to the processes of structural formation and, simultaneously, destruction due to mechanical, thermobaric, and chemical destruction under varying flow conditions. The proposed solution is based on the Verhulst differential equation and tested on five values of shear rates with the addition of correction factors. The article explains the processes of gel formation and the destruction of polymer compounds, and compares the equation of limited growth with the Kenneth Sorbie method, which is employed in the PC-GEL simulator. The limitations of the modern gel invasion model used in the UTCHEM and BPOPE simulator within the porous media (within a narrow gap) are revealed.<\/jats:p>","DOI":"10.3390\/gels12030204","type":"journal-article","created":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:24:34Z","timestamp":1772447074000},"page":"204","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Investigation of the Gelation Process of a Polymer Composition Based on an Acrylic Polymer"],"prefix":"10.3390","volume":"12","author":[{"given":"Inzir","family":"Raupov","sequence":"first","affiliation":[{"name":"Department of Oil and Gas Fields Development and Operation, Empress Catherine II Saint Petersburg Mining University, 199106 Saint Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tatiana","family":"Nosenko","sequence":"additional","affiliation":[{"name":"Chemical Engineering Centre, ITMO University, 197101 Saint Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3039-1925","authenticated-orcid":false,"given":"Victoria","family":"Grigoreva","sequence":"additional","affiliation":[{"name":"Department of Oil and Gas Fields Development and Operation, Empress Catherine II Saint Petersburg Mining University, 199106 Saint Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8135-4801","authenticated-orcid":false,"given":"Vasiliy","family":"Zazulya","sequence":"additional","affiliation":[{"name":"Department of Oil and Gas Fields Development and Operation, Empress Catherine II Saint Petersburg Mining University, 199106 Saint Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gennadiy","family":"Sukhoroslov","sequence":"additional","affiliation":[{"name":"Department of Oil and Gas Fields Development and Operation, Empress Catherine II Saint Petersburg Mining University, 199106 Saint Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vyacheslav","family":"Shkodkin","sequence":"additional","affiliation":[{"name":"Department of Oil and Gas Fields Development and Operation, Empress Catherine II Saint Petersburg Mining University, 199106 Saint Petersburg, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,28]]},"reference":[{"key":"ref_1","first-page":"95","article-title":"Assessment of the Role of the State in the Management of Mineral Resources","volume":"259","author":"Litvinenko","year":"2023","journal-title":"J. Min. Inst."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Palyanitsina, A., Safiullina, E., Byazrov, R., Podoprigora, D., and Alekseenko, A. (2022). Environmentally Safe Technology to Increase Efficiency of High-Viscosity Oil Production for the Objects with Advanced Water Cut. Energies, 15.","DOI":"10.3390\/en15030753"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mardashov, D., Duryagin, V., and Islamov, S. (2021). Technology for Improving the Efficiency of Fractured Reservoir Development Using Gel-Forming Compositions. Energies, 14.","DOI":"10.3390\/en14248254"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"8","DOI":"10.46690\/ager.2024.04.02","article-title":"A Novel Integrated Methodology for Screening, Assessment and Ranking of Promising Oilfields for Polymer Floods","volume":"12","author":"Podoprigora","year":"2024","journal-title":"Adv. Geo-Energy Res."},{"key":"ref_5","first-page":"1281","article-title":"Behavior of solutions of hydrolyzed polyacrylamide in shear and converging flows","volume":"36","author":"Brestkin","year":"1994","journal-title":"Polym. Sci. Ser. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1039\/C9EW00530G","article-title":"Mechanical Degradation of Polyacrylamide at Ultra-High Deformation Rates during Hydraulic Fracturing","volume":"6","author":"Xiong","year":"2020","journal-title":"Environ. Sci. Water Res. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Barabanov, V.L., Demyanovsky, V.B., and Kaushansky, D.A. (2016). The study of rheological heterogeneity of the liquid systems in the instance of the water-swollen dispersed gels of polyacrylamide. Georesour. Geoenergy Geopolit., 4, (In Russian).","DOI":"10.29222\/ipng.2078-5712.2016-13.art4"},{"key":"ref_8","first-page":"127","article-title":"X-ray analysis of conformance control technology based on preformed particle gel treatment","volume":"331","author":"Galkin","year":"2020","journal-title":"Bull. Tomsk. Polytech. Univ. Geo Assets Eng."},{"key":"ref_9","first-page":"251","article-title":"Analysis of the efficiency of waterflooding of oil reservoirs using water-soluble polyacrylamide and preliminary cross-linked polyacrylamide particles","volume":"19","author":"Ketova","year":"2019","journal-title":"Perm J. Pet. Min. Eng."},{"key":"ref_10","first-page":"74","article-title":"The experience of implementation of polymer flooding technology in oil fields characterized by a high degree of salinity of reservoir and injected waters","volume":"4","author":"Mikhailov","year":"2019","journal-title":"Oil Ind."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"388","DOI":"10.2118\/129908-PA","article-title":"Preformed-Particle-Gel Transport Through Open Fractures and Its Effect on Water Flow","volume":"16","author":"Zhang","year":"2011","journal-title":"SPE J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1080\/01932691.2014.1000463","article-title":"Mechanism and Influencing Factors on the Initial Particle Size and Swelling Capability of Viscoelastic Microspheres","volume":"36","author":"Yang","year":"2015","journal-title":"J. Dispers. Sci. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.petsci.2021.09.041","article-title":"A Comprehensive Review of In-Situ Polymer Gel Simulation for Conformance Control","volume":"19","author":"Bai","year":"2022","journal-title":"Pet. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"454","DOI":"10.31897\/pmi.2020.4.8","article-title":"Development of the drilling mud composition for directional wellbore drilling considering rheological parameters of the fluid","volume":"244","author":"Ulyasheva","year":"2020","journal-title":"J. Min. Inst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"848","DOI":"10.5937\/jaes0-31315","article-title":"Application of the CL-Systems Technology for Water Injection Wells at an Oil and Gas Field","volume":"19","author":"Saychenko","year":"2021","journal-title":"J. Appl. Eng. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"232","DOI":"10.29222\/ipng.2078-5712.2024-15-3.art3","article-title":"Description of Nonlinear Viscous and Viscoelastic Properties of Polyacrylamide Solution under Loading","volume":"15","author":"Iktisanov","year":"2024","journal-title":"Actual Probl. Oil Gas"},{"key":"ref_17","unstructured":"Bourgeois, M., and Combe, J. (1981). Basic Rheological Behavior of Xanthan Polysaccharide Solutions in Porous Media: Effects of Pore Size and Polymer Concentration. Enhanced Oil Recovery: Proceedings of the Third European Symposium on Enhanced Oil Recovery, \u00c9ditions Technip."},{"key":"ref_18","first-page":"249","article-title":"Features of Rheological Properties of Polyacrylamide Gel during Loading and Unloading","volume":"39","author":"Iktisanov","year":"2024","journal-title":"Neft. Provintsiya"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Duryagin, V., Nguyen, V.T., Onegov, N., and Shamsutdinova, G. (2023). Investigation of the Selectivity of the Water Shutoff Technology. Energies, 16.","DOI":"10.3390\/en16010366"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"859","DOI":"10.2516\/ogst\/2009057","article-title":"Applied Technologies and Prospects of Conformance Control Treatment in China","volume":"65","author":"Liu","year":"2010","journal-title":"Oil Gas Sci. Technol."},{"key":"ref_21","unstructured":"Ishkov, A.A. (2021, January 14\u201319). Modeling the injection of gelling systems taking into account changes in their rheological properties. Proceedings of the Actual Problems of the Oil and Gas Industry: Proceedings of Three Scientific and Practical Conferences of the Journal \u201cOil Industry\u201d, Moscow, Russia. (In Russian)."},{"key":"ref_22","first-page":"1180","article-title":"Rheological Behavior of Xanthan Gum as a Polymeric Profile Modifier for Polymer Flooding Purposes","volume":"38","author":"Dengaev","year":"2025","journal-title":"Int. J. Eng. Trans. B Appl."},{"key":"ref_23","first-page":"493","article-title":"The Results of the Complex Rheological Studies of the Crosslinked Polymer Composition and the Grounding of Its Injection Volume","volume":"10","author":"Raupov","year":"2019","journal-title":"Int. J. Civ. Eng. Technol."},{"key":"ref_24","first-page":"55","article-title":"Analysis of experience in the use of preformed particle polymer gels in the development of high-water-cut production facilities in low-temperature oil reservoirs","volume":"265","author":"Galkin","year":"2024","journal-title":"J. Min. Inst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"91","DOI":"10.31897\/pmi.2020.1.91","article-title":"Testing of preformed particles polymer gel technology on core filtration models to limit water inflows","volume":"241","author":"Ketova","year":"2020","journal-title":"J. Min. Inst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1016\/j.petrol.2014.06.034","article-title":"A Critical Review on Use of Polymer Microgels for Conformance Control Purposes","volume":"122","author":"Abdilbaki","year":"2014","journal-title":"J. Pet. Sci. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Almohsin, A., Ding, H., and Bai, B. (2018, January 26\u201328). Experimental Study on the Transport and Improved Oil Recovery Mechanism of Submicron Particle Gel. Proceedings of the SPE EOR Conference at Oil and Gas West Asia, Muscat, Oman. SPE-190364-MS.","DOI":"10.2118\/190364-MS"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.2118\/109682-PA","article-title":"Key Aspects of Project Design for Polymer Flooding at the Daqing Oilfield","volume":"11","author":"Wang","year":"2008","journal-title":"SPE Reserv. Eval. Eng."},{"key":"ref_29","unstructured":"Cotton, F.A., Wilkinson, G., Murillo, C.A., and Bochmann, M. (1999). Advanced Inorganic Chemistry, Wiley. [6th ed.]."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1558","DOI":"10.1016\/j.progpolymsci.2011.05.006","article-title":"Polymers for Enhanced Oil Recovery: A Paradigm for Structure\u2013Property Relationship in Aqueous Solution","volume":"36","author":"Wever","year":"2011","journal-title":"Prog. Polym. Sci."},{"key":"ref_31","first-page":"716","article-title":"A study on the mechanism of the gelation reaction of chromium(III) oligomers with polyacrylamide","volume":"52","author":"Han","year":"2001","journal-title":"Acta Polym. Sin."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"59","DOI":"10.2118\/35351-PA","article-title":"Use of preformed gels for conformance control in fractured systems","volume":"12","author":"Seright","year":"1997","journal-title":"SPE Prod. Facil."},{"key":"ref_33","unstructured":"Flory, P.J. (1953). Principles of Polymer Chemistry, Cornell University Press."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Rubinstein, M., and Colby, R.H. (2003). Polymer Physics, Oxford University Press.","DOI":"10.1093\/oso\/9780198520597.001.0001"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Malkin, A.Y., and Kulichikhin, S.G. (1996). Rheokinetics, H\u00fcthig & Wepf.","DOI":"10.1002\/9783527614950"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Delshad, M., Kim, D.H., Magbagbeola, O.A., Huh, C., Pope, G.A., and Tarahhom, F. (2008, January 24\u201328). Mechanistic Interpretation and Utilization of Viscoelastic Behavior of Polymer Solutions for Improved Polymer-Flood Efficiency. Proceedings of the SPE Improved Oil Recovery Symposium, Tulsa, OK, USA. SPE-113620-MS.","DOI":"10.2118\/113620-MS"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.fuel.2018.06.030","article-title":"Novel Viscoelastic Model for Predicting the Synthetic Polymer\u2019s Viscoelastic Behavior in Porous Media Using Direct Extensional Rheological Measurements","volume":"235","author":"Azad","year":"2019","journal-title":"Fuel"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Xiao, K., Mu, L., Wu, X., Yang, L., Xu, F., Zhu, G., and Zhu, Y. (2016, January 21\u201323). Comprehensive Study of Polymer Gel Profile Control for Wag Process in Fractured Reservoir: Using Experimental and Numerical Simulation. Proceedings of the SPE EOR Conference at Oil and Gas West Asia, Muscat, Oman. SPE-179860-MS.","DOI":"10.2118\/179860-MS"},{"key":"ref_39","first-page":"247","article-title":"Incremental-Oil Success from Waterflood Sweep Improvement in Alaska","volume":"25","author":"Ohms","year":"2010","journal-title":"SPE Prod. Oper."},{"key":"ref_40","first-page":"98","article-title":"Using polymeric agents for EOR and waterproofing","volume":"273","author":"Shuvalov","year":"2013","journal-title":"Proc. Gubkin Univ."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Elsharafi, M.O., and Bai, B. (2013, January 10\u201313). Effect of Strong Preformed Particle Gel on Unswept Oil Zones\/Areas During Conformance Control Treatments. Proceedings of the EAGE Annual Conference & Exhibition Incorporating SPE Europec, London, UK. SPE-164879-MS.","DOI":"10.2118\/164879-MS"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1134\/S1061933X19020157","article-title":"Auto-Oscillations in Shear Flow of Colloidal and Polymer Systems: Reality and Instrumental Effects","volume":"81","author":"Skvortsov","year":"2019","journal-title":"Colloid J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.fuel.2015.01.022","article-title":"Optimizing the Strength and Size of Preformed Particle Gels for Better Conformance Control Treatment","volume":"148","author":"Imqam","year":"2015","journal-title":"Fuel"},{"key":"ref_44","first-page":"1343","article-title":"Analysis of Nucleation Time of Gas Hydrates in Presence of Paraffin During Mechanized Oil Production","volume":"37","author":"Korobov","year":"2024","journal-title":"Int. J. Eng. Trans. A Basics."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Elsharafi, M.O., and Bai, B. (2015, January 11\u201313). Minimizing Formation Damage for Preformed Particle Gels in Mature Reservoirs. Proceedings of the SPE Asia Pacific Enhanced Oil Recovery Conference, Kuala Lumpur, Malaysia. SPE-174645-MS.","DOI":"10.2118\/174645-MS"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"65","DOI":"10.2516\/ogst\/2018062","article-title":"Enhanced Oil Recovery Ability of Branched Preformed Particle Gel in Heterogeneous Reservoirs","volume":"73","author":"Yu","year":"2018","journal-title":"Oil Gas Sci. Technol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.colsurfa.2015.03.042","article-title":"Conformation and Plugging Properties of Crosslinked Polymer Microspheres for Profile Control","volume":"477","author":"Lin","year":"2015","journal-title":"Colloids Surf. A Physicochem. Eng. Asp."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"5092","DOI":"10.1021\/ef300689c","article-title":"Selectivity of Pore-Scale Elastic Microspheres as a Novel Profile Control and Oil Displacement Agent","volume":"26","author":"Yao","year":"2012","journal-title":"Energy Fuels"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.jngse.2016.09.026","article-title":"Pressure-Buildup Analysis Method for a Post-Treatment Evaluation of Hydraulically Fractured Tight Gas Wells","volume":"35","author":"Wang","year":"2016","journal-title":"J. Nat. Gas Sci. Eng."},{"key":"ref_50","first-page":"337","article-title":"Analysis of Factors Influencing Mobility and Adsorption in the Flow of Polymer Solution through Porous Media","volume":"14","author":"Hirasaki","year":"1974","journal-title":"SPE J."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1016\/S1876-3804(15)30045-8","article-title":"A Comprehensive Review of Polyacrylamide Polymer Gels for Conformance Control","volume":"42","author":"Bai","year":"2015","journal-title":"Pet. Explor. Dev."},{"key":"ref_52","unstructured":"Schramm, G. (2003). Basics of Practical Rheology and Rheometry, KolosS. (In Russian)."},{"key":"ref_53","first-page":"475","article-title":"Development of Water-Shutoff Composition on the Basis of Carboxymethyl Cellulose for Fractured and Fractured-Porous Oil and Gas Reservoirs","volume":"63","author":"Shagiakhmetov","year":"2018","journal-title":"Acta Tech. CSAV"}],"container-title":["Gels"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2310-2861\/12\/3\/204\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T05:35:36Z","timestamp":1772688936000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2310-2861\/12\/3\/204"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,28]]},"references-count":53,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["gels12030204"],"URL":"https:\/\/doi.org\/10.3390\/gels12030204","relation":{},"ISSN":["2310-2861"],"issn-type":[{"value":"2310-2861","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,28]]}}}