Finite Analytic Numerical Method For Single-phase Steady Flow In Porous Media With Permeability In Tensor Form On 2D Unstructured Grids

Single-phase steady flow in porous media can be described by the quasi-Laplace equation.How to solve this equation accurately is one of the fundamental problem in numerical reservoir simulation.Petroleum reservoirs usually have strong heterogeneities and it brings extra difficulty in numerical simulation.When solving the single-phase steady flow in strong heterogeneous porous media,the traditional numerical methods have to take some average form of the adjacent grids scalar permeabilities as the internodal transmissibility,such as harmonic average,algebraic average,geometric average or others.Unfortunately,these traditional algorithms are all lack of universality.The velocity and the pressure gradient tend to infinity when approaching the singular point joining the different permeability grids.This is why the traditional algorithms fail.When considering the permeability in tensor form,the traditional methods inherit the same idea of simple averaging,and also lead to poor accuracy.The reason is the same as in the scalar permeability case.The above research work was done in the rectangular grids.In this thesis,it has been extended to 2D unstructured grids,and the corresponding finite analytic numerical method has been constructed with permeability in tensor form.In this thesis,considering the angular domain with arbitrary shape,we first derived and obtained the local power-law analytic solution of the quasi-Laplace equation describing the fluid flow in heterogeneous porous media with full tensor permeability.The value of the power exponent of the analytical solution is intrinsic,not affected by the outer boundary conditions,and determined by the permeability distribution of the angular domain.Three basic flow patterns are found:power-law flow,linear flow,and stagnant flow.Different flow patterns can co-exist in the same angular domain,and the analytic solution is just the linear combination of the corresponding basic solutions of these flow patterns.Based on the local analytic solution,the finite analytic numerical method for solving the 2D single-phase steady flow with full tensor permeability on unstructured grids is constructed.Numerical examples show that this numerical method is very accurate,and the convergence speed to the true solution as the grids refining is much faster than the traditional method.More important,the accuracy of FAM doesn't depend on the strength of heterogeneity.In practical applications,the value of refinement parameter n=2 or n = 3 is recommended,and the relative error of the simulation results is below 4%.In contrast,when using the traditional methods,in order to obtain an accurate result,the degree of grids-refinement ratio is required to increase dramatically as the strengthof heterogeneity increases.Furthermore,the analytic solution of the 2D single-phase flow with full tensor permeability in heterogeneous porous media has been obtained,which can be described as an infinite power-law series with a series of intrinsic power exponents for the neighborhood of each singular point.Combined the given boundary conditions,the coefficients of each power law item can be calculated by numerical methods,and then the accurate solution is obtained.Both the finite analytic numerical method proposed and the analytic solution obtained in this thesis focus on the 2D single-phase steady flow in heterogeneous porous media.Actually,since the problems in other physical field,such as thermal conduction,electrostatic fields or others,satisfy the same quasi-Laplace equation,the research achievements of this thesis can be applied directly to these more widely physical fields.