МОДЕЛИРОВАНИЕ ДИНАМИКИ ЧАСТИЧНО НАСЫЩЕННЫХ ПОРОУПРУГИХ ТЕЛ НА ОСНОВЕ МЕТОДА ГРАНИЧНО-ВРЕМЕННЫХ ЭЛЕМЕНТОВ

Abstract

The paper describes the mathematical model for partially saturated porous media based on the Biot’s model with five basic functions to characterize wave process. The mathematical model of the boundary value problem for the three-dimension dynamic theory of poroelasticity is given in Laplace transform. On the basis of the operational calculus theorem about the original integration, the step method of numerical inversion of the Laplace transform is presented. The direct method of boundary integral equations is selected to solve value problems of the three-dimensional dynamic poroelasticity theory, the corresponding boundary integral equation is given. The corresponding matrices of fundamental and singular solutions of the three-dimensional dynamic poroelastic theory are given. A brief description of the boundary-element discretization is presented. Methodological assurance is based on the regularized boundary integral equation usage. The regularized boundary integral equations are written considering the problem of symmetry. The boundary surface of the investigated solid is divided by generalized eight-node quadrangular elements. The consistent elementwise approximation is used. Collocation solution points of the boundary integral equation coincide with the interpolation nodes of unknown boundary functions. To increase the integration accuracy for an element not containing the collocation point, the hierarchical integration algorithm and Gauss integration formulas are applied. Arising discrete analogs are solved by the Gauss-based step process to obtain the values of the boundary functions. Step process is determined by a step algorithm of the numerical Laplace transform inversion. The problem of a unit surface force shock on the free end of the prismatic partially saturated poroelastic solid is considered. Sandstone is selected as the porous material. An analytical solution of the corresponding one-dimensional problem is used for the boundary element model verification. The dependence of computational grid on convergence of the problem solution is investigated; and the impact of step scheme parameter on the solution is examined.