A New Golden Age of Minimal Surfaces

Pérez, Joaquín

doi:10.1090/noti1500

Cited by 21 publications

(16 citation statements)

References 7 publications

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“…While this has been seen in water-wet media [15], for mixed-wet media the unchallenged assumption has been that menisci would change smoothly from spherical interfaces with oil bulging into water (positive capillary pressure) to water bulging into oil (a negative capillary pressure) with the unstated implication that a flat interface would be seen at the transition from positive to negative capillary pressures. Despite the huge interest in other scientific communities in interfaces with curvatures of different sign in orthogonal directions [33], this possibility had been overlooked in the context of porous media flow. As we demonstrate in this paper, in fact such interfaces are likely to be universal in natural systems.…”

Section: Introductionmentioning

confidence: 99%

Minimal surfaces in porous media: Pore-scale imaging of multiphase flow in an altered-wettability Bentheimer sandstone

et al. 2019

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High-resolution x-ray imaging was used in combination with differential pressure measurements to measure relative permeability and capillary pressure simultaneously during a steady-state waterflood experiment on a sample of Bentheimer sandstone 51.6 mm long and 6.1 mm in diameter. After prolonged contact with crude oil to alter the surface wettability, a refined oil and formation brine were injected through the sample at a fixed total flow rate but in a sequence of increasing brine fractional flows. When the pressure across the system stabilized, x-ray tomographic images were taken. The images were used to compute saturation, interfacial area, curvature, and contact angle. From this information relative permeability and capillary pressure were determined as functions of saturation. We compare our results with a previously published experiment under water-wet conditions. The oil relative permeability was lower than in the water-wet case, although a smaller residual oil saturation, of approximately 0.11, was obtained, since the oil remained connected in layers in the altered wettability rock. The capillary pressure was slightly negative and 10 times smaller in magnitude than for the water-wet rock, and approximately constant over a wide range of intermediate saturation. The oil-brine interfacial area was also largely constant in this saturation range. The measured static contact angles had an average of 80 • with a standard deviation of 17 •. We observed that the oil-brine interfaces were not flat, as may be expected for a very low mean curvature, but had two approximately equal, but opposite, curvatures in orthogonal directions. These interfaces were approximately minimal surfaces, which implies well-connected phases. Saddle-shaped menisci swept through the pore space at a constant capillary pressure and with an almost fixed area, removing most of the oil.

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Section: Introductionmentioning

confidence: 99%

Minimal surfaces in porous media: Pore-scale imaging of multiphase flow in an altered-wettability Bentheimer sandstone

et al. 2019

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“…There are several other important aspects of the classical theory of minimal surfaces in Euclidean spaces where substantial progress has been made in recent years but are not covered in this paper; see in particular the survey by Pérez [100] and the monographs by Meeks and Pérez [86,87].…”

Section: Introductionmentioning

confidence: 99%

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

Alarcón¹,

Forstnerič²

2018

J. Aust. Math. Soc.

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In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann-Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi-Yau problem, constructions of properly immersed and embedded minimal surfaces in R n and in minimally convex domains of R n , results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

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“…Every soap film is a physical model of a minimal surface, as the surface tension tries to shrink the surface as much as possible; catenoids belong to the family of minimal surfaces: a minimal surface can be defined to be one where the mean curvature vanishes identically (Oprea, 2000;Pérez, 2017).…”

Section: Minimal Surfaces and Catenoidsmentioning

confidence: 99%

The distortion of a horizontal soap film due to the impact of a falling sphere

Chen

Perera

Jackson

et al. 2019

Chemical Engineering Science

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A horizontal soap film is established in vertical tube a few centimetres in diameter. A metal sphere, 1-2mm diameter, is dropped onto the film, whose distortion is observed by means of a high speed camera. The film wraps partly around the sphere, detaching at a circle which moves up the sphere as it falls. The shape of the film at successive radii, bigger than the radius of contact, was predicted from theory relying on the proposition that if both sides of the film are open to atmosphere, there can be no pressure difference across it. The pressure difference across a film is proportional to (surface tension) / (radius of curvature); hence it follows that the radii of curvature in two planes, perpendicular to each other and to the film surface, must be equal and opposite. This proposition gives equations predicting the shape, in reasonable agreement with experiment. This theory is compared with the theory of catenoids, first studied by Euler in 1744. Catenoid theory gives exactly the same results as the 'radius of curvature' theory presented here. A simple energy conservation argument shows that the two theories are compatible and agree with a published photograph of a soap film catenoid.

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A New Golden Age of Minimal Surfaces

Abstract: Figure 1. A minimal surface, like this soap film, is characterized by the property that small pieces minimize area for given boundary.

Cited by 21 publications

References 7 publications

Minimal surfaces in porous media: Pore-scale imaging of multiphase flow in an altered-wettability Bentheimer sandstone

Minimal surfaces in porous media: Pore-scale imaging of multiphase flow in an altered-wettability Bentheimer sandstone

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

The distortion of a horizontal soap film due to the impact of a falling sphere

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