Quasi-positive families of flux continuous finite volumes schemes in two and three dimensions

• Main Author: Zheng, Hongwen
• Format: Dissertation
• Language: English
• Published: Swansea University 2010

In this thesis, new families of full pressure support flux-continuous, locally conservative, finite-volume schemes are presented for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids in two and three dimensions. The families of flux-continuous schemes have also been referred to in the literature as Multi-point Flux Approximation or MPFA schemes. The schemes are applicable to the general tensor pressure equation with discontinuous coefficients and remove the 0(1) errors introduced by standard reservoir simulation (two-point flux) schemes when applied to full, anisotropic and asymmetric permeability tensor flow approximation. Such tensors may arise when fine scale permeability distributions are upscaled to obtain gridblock-scale permeability distributions. In contrast to the previous MPFA schemes which assume point-wise pressure and flux continuity locally, the new families of schemes presented in the work recover full pressure continuity across the interface between neighboring subcells. The M-matrix conditions [1, 2] define the upper limits for ensuring a local maximum principle is obtained for full-tensor fields. A key condition is that the modulus of the off-diagonal tensor coefficients are bounded by the minimum of the diagonal coefficients. For higher anisotropic ratios, when the resulting discrete matrices violate these bounds these schemes can violate the maximum principle (as with more standard methods) and the numerical pressure solutions can consequently exhibit spurious oscillations. The new family of schemes yield improved performance for challenging problems where earlier flux-continuous schemes exhibit strong spurious oscillations. The M- matrix analysis leads to an optimal quadrature range for these methods. The degree of freedom within the family of full pressure continuity schemes presented is shown to maximise the quadrature range of the flux-continuous schemes. For strongly anisotropic full-tensor cases where M-matrix conditions are violated, it is shown that the earlier families of schemes cannot avoid decoupling of the solution which leads to severe spurious oscillations in the discrete solution. The full quadrature range of the new schemes permits use of quadrature points that were previously out of range for the earlier methods, and that the resulting schemes minimize spurious oscillations in discrete pressure solutions. The new formulation leads to a more robust quasi-positive family of flux-continuous schemes applicable to general discontinuous full-tensor fields. This work also extends the single parameter family of FPS schemes to double families of schemes with general flexibility in quadrature that allow different quadrature points to be used on different control-volume subfaces. The new schemes minimize spurious oscillations in discrete pressure solutions. The new formulation leads to more robust quasi-positive families of flux-continuous schemes applicable to general discontinuous full-tensor fields. The full pressure support flux continuous schemes also extend to 3D on structured and unstructured grids. Surface auxiliary control volume and volume auxiliary control volume are introduced to handle extra degrees of freedom which are required for full pressure continuity over neighboring subcell surface. The new schemes are shown to be beneficial in high anisotropic test cases while remaining comparable with previous tetrahedral pressure support (TPS) schemes in terms of convergence rate. Multi-family schemes in 3D are also presented in this work. This is the extension of 2D double family to 3D. Compared to single family FPS schemes, multi-family schemes are shown to be able to maximize the quadrature and have incomparable flexibility over previous schemes, leading to improved solutions.