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This paper presents a detailed analysis of a numerical solution of three-dimensional transient Darcy flow. The numerical solution of the governing parabolic partial differential equations is obtained by using stabilized mixed Galerkin method and backward Euler method for the discretization of space and time, respectively. The resulting well-posed system of algebraic equations is subsequently solved using conjugate gradient method. The proposed model is validated against Mongan’s analytical model for underground water flow using a set of hexahedral and tetrahedral meshes. The model is used to analyze the transient behavior by simulating the Darcy flow through homogeneous and heterogeneous as well as isotropic and anisotropic media. For large meshes, a parallel algorithm of the transient Darcy flow is also developed for shared memory architecture using OpenMP library. For structured meshes, a speedup of over 22 is obtained on dual AMD Opteron processors. The proposed numerical method for transient Darcy flow offers stability, ease of implementation in higher dimensions and parallel solution for large and complex geometry using standard finite element spaces. [ABSTRACT FROM AUTHOR]

The natural flows through porous media often exhibit transient behavior. Some of the examples include water movement in aquifers, oil flow in reservoirs and blood passing through arteries walls. For accurate modeling of such flows, the Darcy model is used with an additional time-dependent pressure term. In this paper, validation of the three-dimensional numerical solution of transient Darcy flow using the stabilized mixed finite element method is presented. The proposed numerical solution employs the implicit backward difference method for the discretization of time, whereas, for space discretization, the Galerkin technique is used. The model is validated against analytical models including the Theis equation for pressure drawdown near a pumping well. The proposed solution is tested for different values of the viscosity of the fluid, and the permeability and specific storage of the medium. The error analysis shows that the stabilized mixed Galerkin methods give stable solutions with no oscillations and spurious results. It is also found that the viscosity of the fluid and the permeability of the medium have prominent effects on the transient behavior of Darcy flow. [ABSTRACT FROM AUTHOR]

This work deals with the discretization of single-phase Darcy flows in fractured and deformable porous media, including frictional contact at the matrix–fracture interfaces. Fractures are described as a network of planar surfaces leading to so-called mixed-dimensional models. Small displacements and a linear poro-elastic behavior are considered in the matrix. One key difficulty to simulate such coupled poro-mechanical models is related to the formulation and discretization of the contact mechanical sub-problem. Our starting point is based on the mixed formulation using facewise constant Lagrange multipliers along the fractures representing normal and tangential stresses. This is a natural choice for the discretization of the contact dual cone in order to account for complex fracture networks with corners and intersections. It leads to local expressions of the contact conditions and to efficient semi-smooth nonlinear solvers. On the other hand, such a mixed formulation requires to satisfy a compatibility condition between the discrete spaces restricting the choice of the displacement space and potentially leading to sub-optimal accuracy. This motivates the investigation of two alternative formulations based either on a stabilized mixed formulation or on the Nitsche's method. These three types of formulations are first investigated theoretically in order to enhance their connections. Then, they are compared numerically in terms of accuracy and nonlinear convergence. The sensitivity to the choice of the formulation parameters is also investigated. Several 2D test cases are considered with various fracture networks using both P 1 and P 2 conforming Finite Element discretizations of the displacement field and an Hybrid Finite Volume discretization of the mixed-dimensional Darcy flow model. [ABSTRACT FROM AUTHOR]