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Figure C.2: Control-volume , Flux and pressure continuity positions at N, S, E shown on a Quadrilateral.
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In this thesis, families of 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 O(1) errors
introduced by standard reservoir simulation (two-point flux) schemes when applied
to full, anisotropic and asymmetric permeability tensor flow approximation [1, 2, 3].
Such tensors may arise when fine scale permeability distributions are upscaled to
obtain gridblock-scale permeability distributions.
A family of schemes is quantified by a quadrature parametrization where the
position of continuity defines the quadrature and hence the family. In this work
the family of flux-continuous schemes is presented in Physical space and Transform
space, and has been tested for a range of quadrature points. A series of numerical
test cases are presented and a numerical convergence study is conducted for the
family of schemes using different types of two and three dimensional structured and
unstructured grids. Specific quadrature points have been observed to yield improved
convergence for the family of flux-continuous schemes on structured and unstructured
grids in two and three dimensions.
This work also presents a complete extension of the family of control-volume dis-
tributed (CVD) multi-point flux approximation (MPFA) flux-continuous schemes for
general three dimensional grids comprising of different element types e.g., hexahedra,
tetrahedra, prism and pyramid. Discretization principles are presented for each ele-
ment. The pyramid element is shown to be a special case with unique construction
of the continuity conditions. The Darcy flux approximations are applied to a range of test cases that verify consistency of the schemes. Convergence tests of the three-
dimensional families of schemes are presented, with emphasis on use of quadrature
parameterization.
A new family of locally conservative cell-centred flux-continuous schemes is pre-
sented for quadrilateral grids. The new family is defined by introducing a piecewise
constant general geometry-permeability tensor approximation over the subcells of the
control-volumes and ensures that the local discrete general-tensor is elliptic. A fam-
ily of control-volume distributed subcell flux-continuous schemes is defined in terms
of a quadrature parametrization, where the local position of flux continuity defines
the quadrature point and each particular scheme. The subcell tensor approximation
ensures that a symmetric positive definite discretization matrix is obtained for the
base member of the formulation. The physical-space scheme has been shown to be
non-symmetric for general quadrilateral cells [4]. A numerical convergence study of
the schemes shows that the subcell tensor approximation reduces solution errors when
compared to the cell-wise tensor scheme, and the subcell tensor approximation using
the control-volume face geometry yields the best results. A particular quadrature
point is found to improve numerical convergence of the subcell schemes for the cases
tested [5].
When applying the family of flux-continuous schemes to strongly anisotropic het-
erogeneous media they can fail to satisfy a maximum principle (as with other FEM
and finite-volume methods) and result in loss of solution monotonicity for high (full-
tensor) anisotropy ratios causing spurious oscillations in the numerical pressure solu-
tion. In this work methods for obtaining optimal discretization with minimal spurious
oscillations are investigated and the use of flux-splitting techniques [6] is extended to
solve the discrete system for the problems with high anisotropy ratios to improve
solution monotonicity [7, 8]. Flux-splitting schemes are presented together with a
series of numerical results for test-cases with strong anisotropy ratios. In all cases the
resulting numerical pressure solutions are free of spurious oscillations. Monotonicity issues are also discussed and tests performed confirm optimal monotonicity of the
schemes as determined by an M-matrix analysis [2, 3, 9, 10, 11].
This thesis also presents a double (q1, q2)-family of flux continuous schemes. Where,
the double q1, q2-family is quantified by choosing a different quadrature parametriza-
tion or quadrature q on different faces of the sub-cell control-volume leading to a
variable support scheme [10]. M-matrix analysis for double families [10] is then used
to show a key result for general nine-point schemes, which exposes the limits on the
schemes for ensuring solution monotonicity. The analysis is used to determine the
upper limits for obtaining monotonic solutions and to aid the design of schemes that
minimize the occurrence of spurious oscillations in the discrete pressure field.
Finally, the study of a locally conservative family of schemes is applied to upscal-
ing. Equivalent upscaled permeability is used as a measure of performance of the
family of schemes[12]. A series of upscaling examples is presented and convergence
tests are performed for different upscaling techniques. Again the benefits of using the
quadrature parametrization q are highlighted.
A family of flux-continuous, locally conservative, finite-volume schemes has been developed for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids and are control-volume distributed [1,2]. The schemes are applicable to the general tensor pressure equation with discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation (two-point flux) schemes when applied to full tensor flow approximation. The family of flux-continuous schemes is quantified by a quadrature parameterization and has been tested for ranges of quadrature points. Specific points have been observed to yield improved convergence for the family of flux-continuous schemes for structured and unstructured grids in two dimensions [3].
This paper presents a complete extension of the family of control-volume distributed (CVD) multi-point flux approximation (MPFA) flux-continuous schemes for general three dimensional grids comprised of any element type, hexahedra, tetrahedra, prisms and pyramid elements. Discretization principles are presented for each element. The pyramid element is shown to be a special case with unique construction of the continuity conditions. The Darcy flux approximations are applied to a range of test cases that verify consistency of the schemes. Convergence tests of the three-dimensional families of schemes are presented, with emphasis on use of quadrature parameterization. Monotonicity issues are also discussed and tests performed confirm optimal monotonicity of the schemes as determined by an M-matrix analysis [1].
A new family of locally conservative cell-centred flux-continuous schemes is presented for solving the porous media general-tensor pressure equation. A general geometry-permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control-volume distributed subcell flux-continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2:259–290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive-definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical-space schemes are shown to be non-symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non-symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical-space and subcell-space q-families of schemes. M-matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical-space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell-wise constant tensor schemes and that subcell tensor approximation using the control-volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 John Wiley & Sons, Ltd.
