Covering Rational Surfaces with Rational Parametrization Images

Let S be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps f,g,h:A2⇢S⊂Pn such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps f,g˜:A2⇢S, such that the union of its images covers the affine surface S∩An. In the affine case, the number of rational maps involved in the cover is in general optimal.

Four-Dimensional Semi-Riemannian Szabó Manifolds

In this paper, we prove that the deformed Riemannian extension of any affine Szabó manifold is a Szabó pseudo-Riemannian metric and vice versa. We prove that the Ricci tensor of an affine surface is skew-symmetric and nonzero everywhere if and only if the affine surface is Szabó. We also find the necessary and sufficient condition for the affine Szabó surface to be recurrent. We prove that, for an affine Szabó recurrent surface, the recurrence covector of a recurrence tensor is not locally a gradient.

The emergence of small-scale self-affine surface roughness from deformation

Most natural and man-made surfaces appear to be rough on many length scales. There is presently no unifying theory of the origin of roughness or the self-affine nature of surface topography. One likely contributor to the formation of roughness is deformation, which underlies many processes that shape surfaces such as machining, fracture, and wear. Using molecular dynamics, we simulate the biaxial compression of single-crystal Au, the high-entropy alloy Ni36.67Co30Fe16.67Ti16.67, and amorphous Cu50Zr50 and show that even surfaces of homogeneous materials develop a self-affine structure. By characterizing subsurface deformation, we connect the self-affinity of the surface to the spatial correlation of deformation events occurring within the bulk and present scaling relations for the evolution of roughness with strain. These results open routes toward interpreting and engineering roughness profiles.

The hydraulic efficiency of single fractures: Correcting the cubic law parameterization for self-affine surface roughness and fracture closure

Quantifying the hydraulic properties of single fractures is a fundamental requirement to understand fluid flow in fractured reservoirs. For an ideal planar fracture, the effective flow is proportional to the cube of the fracture aperture. Yet, real fractures are rarely planar, and correcting the cubic law in terms of fracture roughness has therefore been a subject of numerous studies in the past. Several empirical relationships between hydraulic and mechanical aperture have been proposed, based upon statistical variations of the aperture field. However, often they exhibit non-unique solutions, attributed to the geometrical variety of naturally occurring fractures. In this study, a non-dimensional fracture roughness quantification-scheme is acquired, opposing effective surface area against relative fracture closure. This is used to capture deviations from the cubic law as a function of quantified fracture roughness, here termed hydraulic efficiencies. For that, we combine existing methods to generate synthetic 3D fracture voxel models. Each fracture consists of two random self-affine surfaces with prescribed roughness amplitude, scaling exponent, and correlation length, which are separated by varying distances to form fracture configurations that are broadly spread in the newly formed two-parameter space. First, we performed a percolation analysis on 600'000 synthetic fractures to narrow down the parameter space on which to conduct fluid flow simulations. This revealed that the fractional amount of contact and the percolation probability solely depends on the relative fracture closure. Next, Stokes flow calculations are performed, using a 3D finite differences code on 6400 fracture models to compute directional permeabilities. The deviations from the cubic law prediction and their statistical variability for equal roughness configurations were quantified. The resulting 2D solution fields reveal decreasing cubic-law accordance's down to 1 % for extreme roughness configurations. We show that the non-uniqueness of the results significantly reduces if the correlation length of the aperture field is much smaller than the spatial extent of the fracture. Under this assumption, an equation was provided that predicts hydraulic efficiencies and respective fracture permeabilities with a mean error of 26.7 %. Numerical inaccuracies were quantified with a resolution test, adding another error of 7 % on top of the previous one. By this, we propose a revised parameterization for the permeability of rough single fractures, which takes numerical inaccuracies of the flow calculations into account. We show that this approach is more accurate, compared to existing formulations. It can be employed to estimate the permeability of fractures if a measure of fracture roughness is available, and it can readily be incorporated in discrete fracture network modeling approaches.