On Solvability of the First Boundary Value Problem for One-Dimensional Internal Erosion
Abstract
This paper deals with a mathematical model of isothermal internal erosion without deformation of a porous medium. Underground water filtration occurs in the aquifer being in contact with frozen sandy soil. During soil thawing and at a certain magnitude of the filtration velocity, soil particles are removed from the flow, and underground cavities are created. These cavities increase in sizes and reach their critical sizes that result in a permafrost arch collapse. A mathematical model is based on mass conservation equations for water, moving solids particles and stationary porous skeleton along with Darcy’s law for water and moving solid particles (similar to a classical Muskat-Leverett model), and the equation for the intensity of suffusion flow. The problem statement and supporting information are provided in Paragraph 1 along with the statement of a theorem of unique classical solvability. Seven lemmas and physical principles for maxima of water saturation and porosity are presented in Paragraph 2. A key moment is to prove Holder’s continuity of saturation. Then, the conditions of Schauder’s theorem of a fixed point are verified.
DOI 10.14258/izvasu(2015)1.2-25
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Vardoulakis I. Sand-production and sand internal erosion: Continuum modeling // Alert School: Geomechanical and Structural Issues in Energy Production. - 2006.
Parron Vera M.A. et al., Analytical solution of coupled soil erosion and consolidation equations by asymptotic expansion approach, Appl. Math. Modell. - 2014.
Кузиков С.С., Папин А.А., Сибин А.Н. Численное моделирование процесса суффозионного выноса грунта // Сборник трудов 17-й регион. конф. по математике «МАК-2014». - Барнаул, 2014.
Кузиков С.С., Папин А.А., Сибин А.Н. Численное исследование профильной задачи внутренней эрозии в межмерзлотном водоносном слое // Известия Алт. гос. ун-та. - 2014. - Вып. 1/2 (85). DOI 10.14258/izvasu(2014)1.2-06.
Папин А.А., Гагарин Л.А., Шепелев В.В., Сибин А.Н., Хворых Д.П. Математическая модель фильтрации грунтовых вод, контактирующих с многолетнемерзлыми породами // Известия Алт. гос. ун-та. - 2013. - Вып. 1/2 (77). DOI 10.14258/izvasu(2013)1.2-06.
Папин А.А., Вайгант В.А., Сибин А.Н. Математическая модель изотермической внутренней эрозии // Известия Алт. гос. ун-та. - 2015. - Вып. 1/1 (85). DOI 10.14258/izvasu(2015)1.1-16.
Антонцев С.Н., Кажихов А.В., Монахов В.Н. Краевые задачи механики неоднородных жидкостей. - Новосибирск, 1983.
Ахмерова И.Г., Папин А.А. Разрешимость краевой задачи для уравнений одномерного движения двухфазной смеси // Математические заметки. - 2014. - Т. 96, №2.
Папин А.А., Ахмерова И.Г. Разрешимость системы уравнений одномерного движения теплопроводной двухфазной смеси // Математические заметки. - 2010. - Т. 87, №2.
Ладыженская О.А., Солонников В.А., Уральцева Н.Н. Линейные и квазилинейные уравнения параболического типа. - М., 1967.
Кружков С.Н., Сукорянский С.М., Краевые задачи для систем уравнений типа двухфазной фильтрации; постановка задач, вопросы разрешимости, обоснование приближенных методов // Матем. сб. - 1977. - Т. 104 (146), №1 (9).
Токарева М.А. Двумерная задача фильтрации в тонком пороупругом слое // Известия Алт. гос. ун-та. - 2013. - Вып. 1/1 (77).
Папин А.А., Токарева М.А. Динамика тающего деформированного снежно-ледового покрова // Вестник Новосибирского государственного университета. - Серия: Математика, механика, информатика. - 2012. - Т. 12, № 4.
Гоман В.А., Папин А.А., Шишмарев К.А. Численное решение двумерной задачи движения воды и воздуха в тающем снеге // Известия Алт. гос. ун-та. - 2014. - Вып. 1/2 (81). DOI 10.14258/ izvasu(2014)1.2-01.
Шишмарев К.А. Тепломассоперенос в тающем снеге // Труды молодых ученых Алт. гос. ун-та. - 2011. - № 8.
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