Application of mixed non-conforming variational formulations for solving the Darcy problem

Предлагается и анализируется построение вычислительной схемы для задачи Дарси с тензорным коэффициентом проницаемости среды при использовании смешанной конечно-элементной аппроксимации на базе разрывного метода Галёркина. Рассматривается построение базиса для скорости в специальном функциональном пространстве Hdiv , а также базиса для давления в пространстве L 2. Проводится верификация полученной вычислительной схемы на классе задач, имеющих аналитическое решение, а также представляются результаты экспериментов с контрастным тензорным коэффициентом проницаемости среды. We consider the construction of a computational scheme for the Darcy model with a tensor permeability coefficient. We use the mixed finite element approximation based on the discontinuous Galerkin formulation. Recently the mixed method has become one of the modern approaches for a numerical solution of the Darcy problem. The main idea of this method is to approximate both the primary and dual variables, while searching for the critical point of the corresponding functional over the finite-element space of admissible test functions, which can be represented as a direct sum of two or more subspaces. This approach permits us to find a solution that corresponds to the physics of the simulated processes. The advantages and disadvantages of the mixed method are discussed in this paper. We use numerical fluxes and jump stabilization following the approach of F. Brezzi, T.J.R. Hughes, L.D. Marini and A. Masud and obtain the computational scheme that is stable at the contrast value of the tensor permeability coefficient. Two hierarchical basis systems for the velocity from

Direct Calculation of Permeability by High-Accurate Finite Difference and Numerical Integration Methods

Velocity of fluid flow in underground porous media is 6~12 orders of magnitudes lower than that in pipelines. If numerical errors are not carefully controlled in this kind of simulations, high distortion of the final results may occur [1–4]. To fit the high accuracy demands of fluid flow simulations in porous media, traditional finite difference methods and numerical integration methods are discussed and corresponding high-accurate methods are developed. When applied to the direct calculation of full-tensor permeability for underground flow, the high-accurate finite difference method is confirmed to have numerical error as low as 10–5% while the high-accurate numerical integration method has numerical error around 0%. Thus, the approach combining the high-accurate finite difference and numerical integration methods is a reliable way to efficiently determine the characteristics of general full-tensor permeability such as maximum and minimum permeability components, principal direction and anisotropic ratio.

Finite Analytic Method for 2D Fluid Flows in Porous Media with Full Tensor Permeability

In this paper, the finite analytic method is developed to solve the two-dimensional fluid flows in heterogeneous porous media with full tensor permeability. With the help of power-law behaviors of pressure and its gradient around the node, a local analytic nodal solution is derived for the pressure equation. Then it is applied to construct a finite analytic numerical scheme which deals with the divergence in pressure gradient. The numerical examples show that the convergence speed of the numerical scheme is fast and independent of the permeability heterogeneity. In contrast, the convergence speed slow rapidly as the heterogeneity increases when the traditional scheme is used.