Nonlinear Evolutionary Problem of Filtration Consolidation With the Non-Classical Conjugation Condition

Abstract

Finite-element solutions of the initial-boundary value problem for a nonlinear parabolic equation in an inhomogeneous domain with the conjugation condition of a non-ideal contact were found. The initial boundary value problem is a mathematical model of an important technical problem of filtration consolidation of inhomogeneous soils. Inhomogeneity is considered in terms of the presence of thin inclusions, physicochemical characteristics of which differ from those of the main soil. The problem of longterm consolidation is especially pronounced in soils with low filtration coefficient. Low permeability of the porous medium causes deviation from the linear relationship between the pressure gradient and the filtration rate. Weak formulation of the problem is suggested, and the accuracy of the approximate finite element solution, its existence and uniqueness are substantiated for the case of Darcy's nonlinear law. A test example and the effect of the nonlinear filtration law for thin inclusion on the dynamics of scattering of excess pressures in the entire area of the problem are considered.

Model of Non-Isothermal Consolidation in the Presence of Geobarriers and the Total Approximation Properties of its Finite Element Solutions

“…We note that according to known field experiments the filtration, thermoosmotic, and thermal conductivity coefficients depend on porosity n. However, in the consolidation problem, as explained in [20], n = n (h). This is taken into account in the problem (3.1)-(3.10).…”

“…Suppose that the functions h 0 (x), T 0 (x) are continuous on each of the closures Ω 1 , Ω 2 . Also regarding the coefficients k, k ω , µ, µ ω , λ, λ ω , c s suppose that 1) In the above definition Ψ h , Ψ T are the sets of functions ψ h (x, t), ψ T (x, t), which together with ∂(•) ∂x , are continuous on each of the closures For further explanations, similarly to the work [20], we note the following. Given condition 1), we get…”

“…It was shown in [6] that the conjugation conditions of non-ideal contact can apply to the contact problems of heterogeneous media even without the presence of fine inclusions. Quite detailed studies of the effect of modified conjugation conditions in the presence of geobarriers on forecast distributions of moisture and excess pressures under the action of chemical and biological (bioclogging) factors were performed in [7,[18][19][20][28][29][30][31][32]. However, problems with the effect of thermal factors have not yet been considered.…”

Balance Boundary Conditions for Thin Boundary Layers: Derivation From the Integral Conjugation Conditions

“…The methodology of deriving these boundary conditions, proposed in this work, allows to take into consideration dependencies of the process parameters along the thickness of the boundary layer. As to the problems with modified conjugation conditions, as described in the proposed methodology, it is demonstrated that significant differences in predictive calculations might arise as compared to the classical conjugation conditions [17,13,10,11]. Our further research will concern numerical experiments and analysis of the influence of these effects on the problems with thin boundary layers.…”

“…Here we propose methodology for deriving boundary conditions in the presence of thin boundary layers based on the integral conjugation conditions for geobarriers. Similar modifications for conjugation conditions with nonlinear dependencies are introduced in works [10,11,12,13]. Further, studies [14,15] adjust the conjugation conditions for thin inclusion to consider the effect of biocolmatation according to the methodology of modelling interconnected processes in porous media described in [16].…”