Higher Dimensional Formulations Research Articles

Higher order cell-based multidimensional upwind schemes for flow in porous media on unstructured grids

Standard reservoir simulation schemes employ single-point upstream weighting for convective flux approximation. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent.New locally conservative cell-based multi-dimensional upwind schemes and higher-order cell-based multi-dimensional upwind schemes that reduce both directional and cross-wind diffusion are presented for convective flow approximation. The new higher-order schemes are comprised of two steps; (a) Higher-order approximation that corrects the directional diffusion of the approximation. (b) Truly multi-dimensional upwind approximation, which involves flux approximation using upwind information obtained by upstream tracing along multi-dimensional flow paths. This approximation reduces cross-wind diffusion. Conditions on tracing direction and CFL number lead to a local maximum principle that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations.Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve a range of unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical cross-wind diffusion effects, leading to improved resolution of concentration and saturation fronts.

Anomalies and time reversal invariance in relativistic hydrodynamics: The second order and higher dimensional formulations

We present two new results on relativistic hydrodynamics with anomalies and external electromagnetic fields, "Chiral MagnetoHydroDynamics" (CMHD). First, we study CMHD in four dimensions at second order in the derivative expansion assuming the conformal/Weyl invariance. We classify all possible independent conformal second order viscous corrections to the energy-momentum tensor and to the U(1) current in the presence of external electric and/or magnetic fields, and identify eighteen terms that originate from the triangle anomaly. We then propose and motivate the following guiding principle to constrain the CMHD: the anomaly--induced terms that are even under the time reversal invariance should not contribute to the local entropy production rate. This allows us to fix thirteen out of the eighteen transport coefficients that enter the second order formulation of CMHD. We also relate one of our second order transport coefficients to the chiral shear waves. Our second subject is hydrodynamics with (N+1)-gon anomaly in an arbitrary 2N dimensions. The effects from the (N+1)-gon anomaly appear in hydrodynamics at (N-1)'th order in the derivative expansion, and we identify precisely N such corrections to the U(1) current. The time reversal constraint is powerful enough to allow us to find the analytic expressions for all transport coefficients. We confirm the validity of our results (and of the proposed guiding principle) by an explicit fluid/gravity computation within the AdS/CFT correspondence.

Higher Order Multidimensional Upwind Convection Schemes for Flow in Porous Media on Structured and Unstructured Quadrilateral Grids

Standard reservoir simulation schemes employ first order upwind schemes for approximation of the convective fluxes when multiple phases or components are present. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent. New locally conservative higher-order multidimensional upwind schemes that significantly reduce both directional and cross-wind diffusion are presented for convective flow approximation. The new schemes are composed of two steps; (a) higher order approximation that corrects the directional diffusion of the approximation and (b) truly multidimensional upwind approximation, which involves flux approximation using upwind information obtained by upstream tracing along wave-vector paths where wave information travels in multiple dimensions. This approximation reduces cross-wind diffusion. Conditions on the tracing direction and Courant–Friedrichs–Levy number lead to a local maximum principle that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations. Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation, including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve structured and unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical cross-wind diffusion effects, leading to improved resolution of concentration and saturation fronts.

Multidimensional upwind convection schemes for flow in porous media on structured and unstructured quadrilateral grids

Standard reservoir simulation schemes employ first order upwind schemes for approximation of the convective fluxes when multiple phases or components are present. These convective flux schemes rely upon upwind information that is determined according to grid geometry. As a consequence directional diffusion is introduced into the solution that is grid dependent. The effect can be particularly important for cases where the flow is across grid coordinate lines and is known as cross-wind diffusion. Truly higher dimensional upwind schemes that minimize cross-wind diffusion are presented for convective flow approximation on quadrilateral unstructured grids. The schemes are locally conservative and yield improved results that are essentially free of spurious oscillations. The higher dimensional schemes are coupled with full tensor Darcy flux approximations. The benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full tensor permeability fields. The test cases involve a range of structured and unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce the numerical cross-wind diffusion effect, leading to improved resolution of concentration and saturation fronts.

High-Dimensional Formulation of a Spinning Particle Interacting with External Gauge Fields

Supersymmetric formulation of a spinning particle interacting with external gauge fields is investigated based on the high-dimensional unification of the gravitational field and non-Abelian gauge fields. It is shown that, associated with the dimensional reduction from the (4+ N)­ dimensional spacetime to the 4-dimensional spacetime and an extra N-dimensional (compact) space, the metric tensor in the extra subspace is absorbed into the mass term of the particle and generates discrete spectrum. Recently, the high-dimensional unified theori),2),3) has renewedly attracted interest as a possible framework including the unified gauge theory of weak and electromagnetic interactions of the Weinberg-Salam type) In such a theory, one firstly sets up a geometry in (4 + N )-dimensional spacetime and then reduces the spacetime into the usual 4-dimensional spacetime and an extra N-dimensional space. In the reduced 4-dimensional theory, there arise the gravitational field, non-Abelian gauge fields and Higgs like scalar fields associated with the com­ ponents of the metric tensor gaP, gai and gij respectively. Here, Greek and Latin letters represent each vector index in the usual spacetime and the extra space. Furthermore, a simple particle in the (4 + N )-dimensional spacetime acquires, after the dimensional reduction, an internal structure as a particle in the usual 4-dimensional spacetime_ In this case, it will be necessary to assume the com­ pactness of the extra space in order to get discrete mass spectra from such an internal structure of the particle. In the high-dimensional unified theory, however, there are various viewpoints on the Higgs like scalar fields. For example, in the 5-dimensional theory of Kaluza and Klein,I) the freedom of such a scalar field was discarded. Scherk and Schwarz 3 ) investigated the possibility of occurring the symmetry breaking of non­ Abelian gauge fields due to the presence of Higgs like scalar fields and showed the instability of vacuum provided that the symmetry group of the extra space is semi­ simple. Thus, in the high-dimensional unified theory, the role of Higgs like scalar fields is not yet decisive. In the present paper, we investigate a supersymmetric formulation of the