Self-similar Solutions for Fluid Displacement Problems in Multilayer Porous Systems
Article Sidebar
Main Article Content
Abstract
To study unsteady filtration and fluid displacement in multilayer porous systems, a self-similar solution has been obtained. From its form, it follows that by a special selection of parameters it is possible to ensure that a certain j-th layer remains isolated throughout the entire process, or that crossflow between layers occurs (for qi > 0 in the case of injection and qi < 0 in the case of withdrawal). Consequently, the form of the obtained solution makes it possible to perform a complete physical analysis of the process under consideration as a whole. The paper considers the problem of injecting one fluid into a system saturated with another fluid. For the problem of fluid displacement in multilayer systems, the existence and uniqueness of the solution are proved, which follows from the monotonicity of the functions Fi(ai) and fi(ai) . It is shown that, in order for the determined unknown parameter to satisfy the condition ai>0, the inequality Pr>P0(i), (i=1,2,3 ..., N) must hold. Therefore, the problem of finding a self-similar solution is solvable only when Qr>Q0.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Gal'tsev YU.V., Gal'tseva O.A. Matematicheskoye modelirovaniye protsessa fil'tratsii zhidkostey v poristoy srede razlichnoy geometrii// nauchnyye vedomosti//Zhurnal: Matematicheskaya fizika, matematicheskoye modelirovaniye. Seriya Matematika. Fizika. 2015 № 23 (220). Vypusk 41.
Ravshanov N., Shadmanova K.U. Issledovaniye matematicheskogo modelirovaniya protsessov fil'tratsii podzemnykh vod v mnogosloynykh neodnorodnykh poristykh sredakh // Mezhdunarodnyy zhurnal teoreticheskikh i prikladnykh voprosov tsifrovykh tekhnologiy. Tom 8 № 1 (2025), 48-56 s. DOI: https://doi.org/10.62132/ijdt.v8i1.232.
Ravshanov N., Abdullayev Z.S., Khafizov O.YA. Modelirovaniye fil'tratsii podzemnykh vod v mnogosloynykh poristykh sredakh// Zhurnal: Stroitel'stvo unikal'nykh zdaniy i sooruzheniy @unistroy. Vyp. 7 (92), 2020, https://sciup.org/143172557DOI: 10.18720/CUBS.92.6.
Brunetti, G.F.A., Maiolo, M., Fallico, C. et al. Unraveling the complexities of a highly heterogene-ous aquifer under convergent radial flow conditions. Engineering with Computers 40, 3115–3130, 2024.
Bubnova O.A., Miroshnyk V.A., Markul R.V., Mashykhina P.B., Tatarko L.H. Mathematical Model-ing of Filtration and Geomigration Under Conditions of Anthropogenic Load. Наука та прогрес транспорту. 2025. № 2. С. 21–28. doi: 10.15802/stp2025/330858.
Lin, J., Ma, R., Sun, Z. et al. Assessing the Connectivity of a Regional Fractured Aquifer Based on a Hydraulic Conductivity Field Reversed by Multi-Well Pumping Tests and Numerical Groundwater Flow Modeling. J. Earth Sci. 34, 1926–1939, 2023. https://doi.org/10.1007/s12583-022-1674-5.
Akingboye, A.S., Bery, A.A., Kayode, J.S. et al. Near-Surface Crustal Architecture and Geohydrody-namics of the Crystalline Basement Terrain of Araromi, Akungba-Akoko, SW Nigeria, Derived from Multi-Geophysical Methods. Nat Resour Res 31, 215–236 (2022).
Jwan Sabah Mustafa, Dana Khider Mawlood, Mathematical modelling for groundwater manage-ment for multilayers aquifers (Erbil basin), Ain Shams Engineering Journal, Volume 15, Issue 7, Pp 1, 2024, 102781, ISSN 2090-4479, https://doi.org/10.1016/j.asej.2024.102781.
Akylas E, Gravanis E. Approximate Solutions of the Boussinesq Equation for Horizontal Uncon-fined Aquifers During Pure Drainage Phase. Water. 2024; Pp 1, 16(20):2984.
Xiong, M., Tong, C., & Huang, C.-S. A new approach to three-dimensional flow in a pumped con-fined aquifer connected to a shallow stream: Near-stream and far-from-stream groundwater extrac-tions. Water Resources Research, 2021, Pp 1. 57, e2020WR028780.
Kamel Al-Khaled, Mohamed Ali Hajji. Mathematical modeling to simulate the movement of con-taminants in groundwater. Journal of Applied Analysis & Computation, 2016, 6(1): Pp.156-170.
Condon, L. E., Kollet, S., Bierkens, M. F. P., Fogg, G. E., Maxwell, R. M., Hill, M. C., et al. (2021). Global groundwater modeling and monitoring: Opportunities and challenges. Water Re-sources Research, 57, e2020WR029500. https://doi.org/10.1029/2020WR029500.
Josie Geris, Jean-Christophe Comte, Fulvio Franchi, Alfred K. Petros, et al., Surface water-groundwater interactions and local land use control water quality impacts of extreme rainfall and flooding in a vulnerable semi-arid region of Sub-Saharan Africa, Journal of Hydrology, Vol. 609, 2022, 127834.
Vabishchevich P.N., Vasil'yev V.I., Vasil'yeva M.V. i Nikiforov D.YA. Chislennoye resheniye odnoy obratnoy zadachi fil'tratsii, //Uchenyye zapiski Kazanskogo universiteta, Seriya, Fiziko-matematicheskiye nauki, -Kazan', (2015), T. 157, №4, -S. 79–89.
Abutaliyev F.B., Nekotoryye analiticheskiye i chislennyye resheniya zadach neustanovivshiyesya techeniy v odnosloynykh i mnogosloynykh plasta, //Diss. na soiskaniye uchenoy stepeni doktora f.-m.n. –T., (1969), 433s.
Pushkov I.N., Matematicheskoye modelirovaniye dinamiki vnedreniya podzemnykh vod pri otrabotke gazovogo mestorozhdeniya, //Gazovaya promyshlennost', -S.51-54.
Korolov V.A. Vzaimosvyaz' potentsiala vlagi v gruntakh s parametrami ispareniya iz nikh vody, //Inzhenernaya Geologiya-201, №3, -S. 22-33.
Sologayev V. I. i Chernov D. A., Modelirovaniye nestatsionarnoy fil'tratsii v napornykh potokakh pri turbulentnom dvizhenii s pomoshch'yu elektronnykh tablits pri proyektirovanii zashchity ot podtopleniya, //Omskiy nauchnyy vestnik, (2014), №2(130), -S. 260-263, https://cyberleninka.ru/article/n/modelirovanie-nestatsionarnoy-filtratsii-v-napornyh-potokah-pri -turbulentnom-dvizhenii-s-pomoschyu-elektronnyh-tablits-pri.
Kosichenko YU.M., Baklanova D.V., Bayev O.A. i Kosichenko M.YU., Raschet podpora gruntovykh vod mezhdurechnogo massiva pri neustanovivsheysya fil'tratsii iz vodokhranilishcha, //Melioratsiya i gidrotekhnika, (2014), №3(15).
Sologayev V.I., O modelirovanii naporno-beznapornoy fil'tratsii vody v gorodskom stroitel'stve, //Nauchnyy retsenziruyemyy zhurnal "Vestnik SibADI", (2017), №2(54), -S.124-128, https://doi.org/10.26518/2071-7296-2017-2(54)-4-30.
Khublaryan M. G., Zyryanov V. N. i Frolov A. P., Nelineynyye aspekty fil'tratsii gruntovykh vod, //Vodnyye resursy, gidrofizicheskiye protsessy, (2010), T. 37, №2, -S. 176-185, https://naukarus.com/nelineynye-aspekty-filtratsii-gruntovyh-vod.