Mathematical modeling of the dynamics of consolidation processes with relaxation effects
- PDF / 124,529 Bytes
- 7 Pages / 595.276 x 793.701 pts Page_size
- 56 Downloads / 177 Views
MATHEMATICAL MODELING OF THE DYNAMICS OF CONSOLIDATION PROCESSES WITH RELAXATION EFFECTS V. M. Bulavatskya and V. V. Skopetskiia†
UDC 517.954: 532.546
The consolidation of salt-saturated porous media is modeled with space–time nonlocality effects and with/without the relaxation of the filtration rate taken into account. A numerical algorithm for modeling the dynamics of the process is proposed and an asymptotic analysis of the problem for excess head is performed for weak spatial nonlocality. Keywords: mathematical modeling, consolidation, mass transfer, relaxation, nonclassical models, systems of partial differential equations, boundary-value problems, asymptotic approximations, numerical solutions. INTRODUCTION Since the anthropogenic impact on the environment intensifies, theoretical study of man-made environmental factors has become especially important. One of the adverse effects of human activity is the pollution of soil and subsoil waters by various chemical substances and industrial wastes. Theoretical studies in this field are of current interest because of not only the practical need for analyzing the conditions of ecologically safe operation of various engineering objects but also the expensive (or sometimes impossible) full-scale modeling of the dynamic processes in these objects. This intensified efforts in mathematical modeling of mass transfer of salt solutions during filtration consolidation of soil foundations of sewage ponds and other hydraulic-engineering objects [1–6]. Note that though methods of mathematical modeling within the framework of this subject have been substantially developed, they are still far from being perfect. For example, the mathematical models of consolidation currently in use disregard some factors that exercise a significant influence on the dynamics of the process under complex mining and geological conditions and in case of a complex microstructure of the salt solutions in sewage ponds. In this connection, it becomes important to take into account the relaxation properties of both the filtration process and deformation of a porous medium [1]. For example, the filtration consolidation of a porous medium saturated with a salt solution is modeled in [2] as a process in a double-relaxation system: relaxation filtration of a pore fluid in a relaxation-compressible medium. The present paper deals with the mathematical modeling of the filtration consolidation of deformable porous media saturated with salt solutions, with allowance for space–time nonlicality effects [7], which allows, in some cases, more accurate modeling of the dynamics of these processes under complex conditions. CONSTRUCTING A MATHEMATICAL MODEL REGARDLESS OF FILTRATION RATE RELAXATION. BOUNDARY-VALUE PROBLEM FORMULATION Let us use the following generalization of Darcy’s law [8, 9] to the motion of salt solutions strongly affected by space–time nonlocality ¶æ ¶H ¶ 2 H ö÷ ¶Ñ , (1) ux = - k ç H + l 1 - l2 ±n 2 ÷ ¶x çè ¶t ¶x ¶x ø a
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukrain
Data Loading...
