Application of the Galerkin's Finite Element Method to the Flow of Power-Law Non-Newtonian Fluids through Porous Media

摘要

Conventional approach of modeling fluid flow through porous media and analysis of pressure data often adopt a simplified approach by assuming a Newtonian fluid flow behavior. However, most of the fluids encountered in the oilfield exhibit non-Newtonian flow characteristics. Analysis of field data involving these fluids with conventional Newtonian flow models would result in erroneous interpretation. The generalized equation for the radial flow of non-Newtonian fluids through porous media has been solved analytically but a numerical solution by finite element is yet to be published. In this study, the Galerkin Finite Element technique is used to solve the generalized problem under the various inner and outer boundary conditions that are frequently encountered in the oilfield. Weak formulation of the generalized PDE and boundary conditions is obtained and is discretized into finite elements. Element matrix in both the spatial and time domains are computed using Gauss quadrature technique. Subsequently, the nodal pressure values are computed and a derivative function is used to compute the corresponding pressure derivatives. A commercial simulator is used to verify the numerical solution for the conventional Newtonian fluid flow. The numerical solution is also compared to approximate analytic solutions involving non-Newtonian fluids. Results show the commonly observed flow regimes in pressure transient testing, which include the earlytime unit slope line (wellbore storage), the early-time trough signifying pseudo-steady state matrix fracture flow (naturally-fractured reservoirs), the infinite-acting radial flow regime line, and the late-time boundarydominated footprints for no-flow boundary, constant-pressure boundary, and pseudo-steady state (closed reservoir system). The solution obtained in this study matches the numerical solution from a commercial simulator. Comparison of the numerical solution with approximate analytic solutions also shows strong agreement. Validation of the finite element solution with field data involving polymer injection shows a satisfactory match. Parametric study of the effect of fluid flow index indicates the strong dependence of pressure transient and characteristic flow regimes on fluid rheology. Limited studies have established the application of finite element technique in modeling non-Newtonian fluid flow through porous media. This study demonstrates how the generalized fluid flow model can be solved by finite element method.