Finite Analytic Numerical Method For Three-dimensional Single-phase Flow In Heterogeneous Porous Media

Numerical simulation is a necessary process during the exploitation of oil and gas in a reservoir. The reservoir is usually heterogeneous and this brings a lot of difficulties for the numerical reservoir simulation. In the traditional numerical methods, the intermodal transmissibility is usually assumed as the harmonic mean value of the two permeabilities of the adjacent grids. Unfortunately, this algorithm underestimates the flux, and its error is out of control. In order to conquer this, a high efficient finite analytic numerical method for the 3D steady single-phase flow in the heterogeneous porous media will be proposed in this dissertation.At first, we consider the case with scalar permeability. A quasi-2D assumption is proposed according to the fact that the pressure gradient along the common edge of the control volume is of limited value, while the pressure gradient normal to the edge will tend to infinity when approaching the edge. So the pressure gradient along the edge can be ignored, and the 3D quasi-Laplace equation will degenerate to a 2D one around each edge of the control volume. With the help of the power-law analytic solution of the 2D quasi-Laplace equation around a singular point, an approximate analytic solution of the 3D quasi-Laplace equation around the edge can be found. Based upon this, a finite analytic method for solving the 3D quasi-Laplace equation is constructed. Numerical examples show that, only with 2x2x2 or 3x3x3 subdivisions, the proposed scheme can provide rather accurate solutions; and its relative error of the calculated average permeability is below 5%. More important, the calculation accuracy of the proposed scheme is independent of the strength of the permeability heterogeneity. In contrast, the calculation efficiency of the traditional method depends on the heterogeneity of the porous media and the error is uncontrolled. The proposed numerical scheme is utilized to test the well-known LLM (Landau, Lifshitz and Matheron) conjecture, which provides keq/kG=exp(1/6×σink2) for the isotropic log-normal porous medium (kG represents the geometric average of the sample permeability; D is the dimension and σInk2 is the variance of the logarithm of permeability). The numerical results does not support this conjecture for large σInk, but strongly suggest the linear relation Keq/kc=1+1/6×σInk2.Secondly, the full tensor permeability case is considered in this dissertation. In this situation, the quasi-two-dimensional assumption still holds. But different from the scalar permeability case, the pressure gradient along the edge will induce the corresponding flux in the plane normal to the edge because the cross component of the permeability tensor is nonzero. If the induced flux is neglected, still with the help of the 2D power-law analytic solution, the finite analytic numerical scheme I can be constructed. This scheme is similar as the one for the sealer permeability case, and still has high accuracy. To further improve the calculation accuracy, the finite analytic numerical scheme Ⅱ is constructed, where the induced flow is considered. Under this situation, the approximate 3D analytic solution around the control volume edge can be expressed as a combination of the quasi-2D power-law solution and the induced linear solution. The finite analytic numerical scheme Ⅱ is a little more complicated but more accurate. The numerical results show that the finite analytic scheme Ⅱ is more accurate than the scheme I. For the scheme I, the relative errors of the calculated flux in the pressure drop direction are below 10% with 4x4x4 subdivisions. For the schemeII,the relative errors are below 8.5% correspondingly. When considering the calculated flux normal to the pressure drop direction, the scheme Ⅱ is still more accurate than the scheme I. With 4x4x4 subdivisions, the maximum relative error of the flux normal to the pressure drop direction is only 6.2%. The finite analytic scheme Ⅰ and Ⅱ are also compared with the traditional MPFA method, and the high calculation efficiency of the proposed schemes is confirmed.The finite analytic scheme proposed in this dissertation can also be applied to the numerical simulation for other physical problems which can be described by the quasi-Laplace equation, such as steady heat transfer, electrostatic field, mass transfer and so on.